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The concept of energy (and the conservation of energy) is one of the most important topics in physics. Work Dot products Energy approach is simpler than Newton’s second law. Chapter 7: Work and Kinetic Energy-1 Reading assignment: Chapter Homework: due Monday, Sept 28, 2009 Chapter 7: 5AE's, 5AF's, Q7, 2, 9, 18, 22

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1. A woman holds a bowling ball in a fixed position. The work she does on the ball ___ 1. depends on the weight of the ball. ___ 2. cannot be calculated without more information. ___ 3. is equal to zero. 2. A man pushes a very heavy load across a horizontal floor. The work done by gravity on the load ___ 1. depends on the weight of the load. ___ 2. cannot be calculated without more information. ___ 3. is equal to zero.

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3. When net positive work is done on a particle, its kinetic energy a. increases. b. decreases. c. remains the same. d. need more information about the way the work was done 4. In a collision between two billiard balls, a. energy is not conserved if the collision is perfectly elastic. b. momentum is not conserved if the collision is inelastic. c. not covered in the reading assignment

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Work (as defined by a physicist) Definition: The work done on an object by an external force is - the product of the component of the force in the direction of the displacement and the magnitude of the displacement.

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How much work is done when just holding up an object?

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How much work is done when the displacement is perpendicular to the force?

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What is the work done by - the gravitational force - the normal force - the force F when the block is displaced along the horizontal. The total work is: - the sum of the work done by all forces - or: d = 10m = 60° F = 10 N

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What is the work done when lifting? - By the gravitational force? - By the applied force? Sign convention: W is positive: If F and d are parallel If energy is transferred into the system W is positive: If F and d are antiparallel If energy is transferred out of the system Strongest man lifting 140 kg boulder by 1 m.

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Work is a scalar quantitiy. (not a vector) Work has units of newton·meter (N·m) = the joule (J)

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A man loads a refrigerator onto a truck using a ramp. Ignore friction. He claims he would be doing less work if the length of the ramp would be longer. Is this true? Black board example 7.5

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Black board example 7.1 A donkey is pulling a cart with a force of magnitude F = 500 N at an angle of 30º with the horizontal. Calculate the work done by the donkey as the cart is pulled for one mile (1648 m). 500 N

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Work is the scalar product (or dot product) of the force F and the displacement d. F and d are vectors W is a scalar quantity Definition of dot product and work

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Scalar product between vector A and B Scalar product is commutative: Definition: Distributive law of multiplication:

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Scalar Product using unit vectors: We have the vectors A and B: Then:

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Black board example 7.2 A particle moving in the x-y plane undergoes a displacement d = (2.0i + 3.0j) m at a constant force F = (5.0i + 2.0j) N acts on the particle. Calculate (a)The magnitude of the displacement and the force. (b)The work done by F. (c)The angle between F and d.

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1. The gravitational potential energy of a particle at a height z above Earth’s surface ___ 1. depends on the height z. ___ 2. depends on the path taken to bring the particle to z. ___ 3. both 1 and 2. ___ 4. is not covered in the reading assignment. 2. Which of the following is not a conservative force? ___ 1. the force exerted by a spring on a particle in one dimension ___ 2. the force of friction ___ 3. the force of gravity ___ 4. not covered in the reading assignment 3. Which of the following was not discussed in the reading assignment? ___ 1. conservation of non-conservative forces ___ 2. block and tackle ___ 3. work ___ 4. all of the above were discussed

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Black board example 7.2 A particle moving in the x-y plane undergoes a displacement d = (2.0i + 3.0j) m at a constant force F = (5.0i + 2.0j) N acts on the particle. Calculate (a)The magnitude of the displacement and the force. (b)The work done by F. (c)The angle between F and d.

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What if the force varies? We have to integrate the force along x Work done by a varying force: Thus, the work is equal to the area under the F(x) vs. x curve.

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Black board example 7.3 A force acting on a particle varies as shown in the Figure. What is the total work done on the particle as it is moved from x = 0 to x = 8 m? Hint: It is the area under the curve.

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Consider a spring Hooke’s law: (Force required to stretch or compress a spring by x): k is the spring constant of a spring. Stiff springs have a large k value.

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Work done by a spring x i x f

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A spring-loaded toy dart gun is used to shoot a dart straight up in the air, and the dart reaches a maximum height of 24 m.The same dart is shot straight up a second time from the same gun, but this time the spring is compressed only half as far before firing. How far up does the dart go this time, neglecting friction and assuming an ideal spring? m m m m 5. 6 m 6. 3 m 7. impossible to determine

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Black board example 7.3 A kg mass is hung from a spring extending the spring by a distance x = 0.2 m (a)What is the spring constant of the spring? (b)How much work was done on the mass by the gravitational force (c)How much work was done on the mass by the spring force?

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The kinetic energy of a particle is:

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A bullet of mass m = kg moves at 500 m/s. A truck of mass m = 1000 kg moves at 5 m/s Which has more kinetic energy?

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Work due to friction If friction is involved in moving objects, work has to be done against the kinetic frictional force. This work is:

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A cart on an air track is moving at 0.5 m/s when the air is suddenly turned off. The cart comes to rest after traveling 1m. The experiment is repeated, but now the cart is moving at 1 m/s when the air is turned off. How far does the cart travel before coming to rest? 1. 1 m 2. 2 m 3. 3 m 4. 4 m 5. 5 m 6. impossible to determine

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Black board example 7.4 Angus is pulling a 10,000 kg truck with all his might (2000N) on a frictionless surface for 10.0 m. (a)How much work is the man doing? (b)What is the speed of the truck after 10 m. (c)What is the speed of the truck after 10 m if there is friction? (friction coefficient: )

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Power Power is the rate at which work is done: Average power: (work done per time interval t)

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The power can also be expressed as: The units of power are joule/sec (J/s) = watt (W) (Dot product)

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Black board example 7.7 An elevator having a total mass of 3000 kg moves upward against the gravitational force at a constant speed of 9.13 m/s. (a)What is the power delivered by the motor?

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One form of energy can be converted into another form of energy. Conservative and non-conservative forces CONSERVATION OF ENERGY Chapter 8: Potential Energy and Conservation of Energy part 1 (finish Chp7) Reading assignment: Chapter Homework :due Monday, October 5, 2009 Problems:Chapter 8 33, 36, 37, 58, 65 Bonus: 48, 64, 37

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1. Suppose you know the potential energy function corresponding to a force. Is it always possible to calculate the force? ___ 1. yes ___ 2. only if the force is nonconservative ___ 3. not covered in the reading assignment 2. The potential energy of a spring is ___ 1. proportional to the amount the spring is stretched. ___ 2. proportional to the square of the amount the spring is stretched. ___ 3. not yet covered in any reading assignment. 3. A car slows down as a result of air friction. Which is true? ___ 1. The car’s kinetic energy decreases. ___ 2. Heat is generated. ___ 3. The energy of the car/road/air system is constant. ___ 4. all of the above ___ 5. none of the above

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Potential energy U: -Can be thought of as stored energy that can either do work or be converted to kinetic energy. -When work gets done on an object, its potential and/or kinetic energy increases. -There are different types of potential energy: 1.Gravitational energy 2.Elastic potential energy (energy in an stretched spring) 3.Others (magnetic, electric, chemical, …)

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Conservative and non-conservative forces Conservative forces: Work is independent of the path taken. Work depends only on the final and initial point. Work done is zero if the path is a closed loop (same beginning and ending points.) We can always associate a potential energy with conservative forces. We can only associate a potential energy with conservative forces. Work done by a conservative force: W c = U i – U f = - U Examples of conservative forces: _____________________________________________

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Conservative forces and potential energy The work done by a conservative force equals the negative of the change in potential energy associated with that force. Any conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x.

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Non-conservative forces: A force is non-conservative if it causes a change in mechanical energy; mechanical energy is the sum of kinetic and potential energy. Example: Frictional force. - This energy cannot be converted back into other forms of energy (irreversible). - Work does depend on path. Conservative and non-conservative forces Sliding a book on a table

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Gravitational potential energy: - Potential energy only depends on y (height) and not on x (lateral distance)

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You are 1.80 m tall. A 0.1 kg apple, which is hanging 1 m above your head, drops on you. What is the difference in gravitational potential energy when it hangs and when it hits you? (a)How much gravitational potential energy does it loose? 1 m Black board example 8.1

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Work done by/on a spring: x i x f Elastic potential energy stored in a spring: The spring is stretched or compresses from its equilibrium position by x

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Review Important energy formulas: Work: Forms of energy:

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Conversion of: Elastic potential energy into kinetic energy into gravitational potential energy Demo example (conversion of energy): (ballistic pendulum)

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A mass m is bobbing up and down on a spring. Describe the various forms of energy of this system. (a)At the highest point (b)At the point where the kinetic energy is highest (c)At the lowest point Black board example 8.2

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Black board example 8.3 Three balls are thrown from the top of a building, all with the same initial speed. The first is thrown horizontally, the second with some angle above the horizontal and the third with some angle below the horizontal. (a)Describe the motion of the balls. (b)Rank the speed of the balls as they hit the ground.

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Black board example 8.5 Nose crusher? A bowling ball of mass m is suspended from the ceiling by a cord of length L. The ball is released from rest when the cord makes an angle A with the vertical. (a)Find the speed of the ball at the lowest point B. (b)Assume a cord length L = 5m and an angle A = 20°. (c)The ball swings back. Will it crush the operator’s nose?

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Reading potential energy curves Remember:

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Black board example 8.6 During a rock slide, a 520 kg rock slides from rest down a hillside that is 500 m long and 300 m high. The coefficient of friction between the rock and the hillside is (a)What is the gravitational potential energy of the rock before the slide? (b)How much energy is transferred into thermal energy during the slide? (c)What is the kinetic energy of the rock as it reaches the bottom of the hill?

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One form of energy can be converted into another form of energy. Conservative and non-conservative forces CONSERVATION OF ENERGY Chapter 8: Potential Energy and Conservation of Energy part 2 Reading assignment: Chapter 9 Homework : (due Wednesday, Oct. 7, 2009): Problems:10AE's, Q2, 3, 6, 19, 20

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(a) Can an object-Earth system have kinetic energy and not gravitational potential energy? Yes No (b) Can it have gravitational potential energy and not kinetic energy? Yes No (c) Can it have both types of energy at the same moment? Yes No (d) Can it have neither? Yes No

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1) A ball of clay falls freely to the hard floor. It does not bounce noticeably, but very quickly comes to rest. What then has happened to the energy the ball had while it was falling? a) Most of it went into sound. b) It has been transformed back into potential energy. c) It is in the ball and floor as energy of invisible molecular motion. d) It has been used up in producing the downward motion. e) It has been transferred into the ball by heat. 2) You hold a slingshot at arm's length, pull the light elastic band back to your chin, and release it to launch a pebble horizontally with speed 100 cm/s. With the same procedure, you fire a bean with speed 500 cm/s. What is the ratio of the mass of the bean to the mass of the pebble (bean/pebble)? a) 1/5 b) 1/√5 c) 1 d) √5 e) 5

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Conservation of mechanical energy If we deal only with conservative forces and If we deal with an isolated system (no energy added or removed): The total mechanical energy of a system remains constant!!!! The final and initial energy of a system remain the same: E i = E f E… total energy K… Kinetic energy U… potential energy Thus:

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Work due to friction If friction is involved in moving objects, work has to be done against the kinetic frictional force. This work is:

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Work done by non-conservative forces 1. Work done by an applied force. (System is not isolated) An applied force can transfer energy into or out of the system. Example. Applying a force to an object and lifting increases the energy of the object.

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Work done by non-conservative forces 2. Situations involving kinetic friction. (Friction is not a conservative force). Kinetic friction is an example of a non-conservative force. If an object moves over a surface through a distance d, and it experiences a kinetic frictional force of f k it is loosing kinetic energy Thus, the mechanical energy (E = U + K) of the system is reduced by this amount.

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A frictionless roller coaster with an initial speed of v 0 = m/s, at the initial height h = m, has a mass m = kg (a)What is the speed at point A? (b)What is the speed at point B (c)How high will it move up on the last hill? Black board example 8.4 HW 12

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3) A pile driver is a device used to drive posts into the Earth by repeatedly dropping a heavy object on them. Assume the object is dropped from the same height each time. By what factor does the energy of the pile driver-Earth system change when the mass of the object being dropped is tripled? a) 1/9 b) 1/3 c) 1: the energy is the same d) 3 e) 9

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A curving children's slide is installed next to a backyard swimming pool. Two children climb to a platform at the top of the slide. The smaller child hops off to jump straight down into the pool and the larger child releases herself at the top of the frictionless slide. (4) Upon reaching the water, how does the kinetic energy of the larger child compare to that of the smaller child? a) greater than b) equal to c) less than (5) Upon reaching the water, how does the speed of the larger child compare to that of the smaller child? a) equal to b) less than c) greater than (6) During the motions from the platform to the water, how does the acceleration of the larger child compare to that of the smaller child? a) equal to b) less than c) greater than

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Black board example 8.7 A 3.5 kg block is accelerated by a compressed spring whose spring constant is 640 N/m. After leaving the spring at the spring’s relaxed length, the block travels over a horizontal surface with a frictional coefficient k = 0.25 for a distance of 7.8 m. (a)What is the increase in the thermal energy of the block-floor system? (b)What is the maximum kinetic energy of the block? (c)Through what distance was the spring compressed initially (before the block moved)?

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Suppose you drop a 1-kg rock from a height of 5 m above the ground. When it hits, how much force does the rock exert on the ground? N 2. 5 N N N 5. impossible to determine

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Center of mass Momentum Momentum is conserved Chapter 9: Linear Momentum & Collisions Reading assignment: Chapter Homework : (due Saturday, Oct. 10, 2009): Problems:Chapter 9: 5AF's, Q11, 28, 32, 40, 68, 69

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1. The impulse delivered to a body by a force is ___ 1. defined only for interactions of short duration. ___ 2. equal to the change in momentum of the body. ___ 3. equal to the area under an F vs. x graph. ___ 4. defined only for elastic collisions. 2. In an elastic collision ___ 1. energy is conserved. ___ 2. momentum is conserved. ___ 3. the magnitude of the relative velocity is conserved. ___ 4. all of the above 3. In an inelastic collision ___ 1. both energy and momentum are conserved. ___ 2. energy is conserved. ___ 3. momentum is conserved. ___ 4. neither is conserved.

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4. In two-dimensional elastic collisions, the conservation laws ___ 1. allow us to determine the final motion. ___ 2. place restrictions on possible final motions. ___ 3. do not allow us to say anything about the final motion. ___ 4. are not covered in the reading assignment.

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Suppose you drop a 1-kg rock from a height of 5 m above the ground. When it hits, how much force does the rock exert on the ground? N 2. 5 N N N 5. impossible to determine

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Center of mass Center of mass for many particles: Where is the center of mass of the arrangement of particles below. (m 3 = 2 kg and m 1 = m 2 = 1 kg)? Black board example 9.1

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A method for finding the center of mass of any object. - Hang object from two or more points. - Draw extension of suspension line. - Center of mass is at intercept of these lines.

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Center of mass of a solid body (uniform density)

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Black board example 9.2 A uniform square plate 6 m on a side has had a square piece 2 m on a side cut out of it. The center of that piece is at x = 2 m, y = 0. The center of the square plate is at x = y = 0. Find the coordinates of the center of mass of the remaining piece.

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Motion of a System of Particles. Newton’s second law for a System of Particles The center of mass of a system of particles (combined mass M) moves like one equivalent particle of mass M would move under the influence of an external force.

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A rocket is shot up in the air and explodes. Describe the motion of the center of mass before and after the explosion.

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The linear momentum of a particle of mass m and velocity v is defined as The linear momentum is a vector quantity. It’s direction is along v. The components of the momentum of a particle: Linear Momentum

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From Newton’s second law: The time rate of change in linear momentum is equal to the net forces acting on the particle. This is also true for a system of particles: Total momentum = Total mass ·velocity of center of mass And: Net external force = rate of change in momentum of the center of mass

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Thus: If no external force is acting on a particle, it’s momentum is conserved. This is also true for a system of particles: If no external forces interact with a system of particles the total momentum of the system remains constant. Conservation of linear momentum

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Suppose you are on a cart, initially at rest on a track with very little friction. You throw balls at a partition that is rigidly mounted on the cart. If the balls bounce straight back as shown in the figure, is the cart put in motion? 1. Yes, it moves to the right. 2. Yes, it moves to the left. 3. No, it remains in place.

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Black board example 9.3 You (100kg) and your skinny friend (50.0 kg) stand face-to-face on a frictionless, frozen pond. You push off each other. You move backwards with a speed of 5.00 m/s. (a)What is the total momentum of the you-and-your-friend system? (b)What is your momentum after you pushed off? (c)What is your friends speed after you pushed off?

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Momentum Momentum is conserved – even in collisions with energy loss due to friction/deformation. Impulse Chapter 9: Linear Momentum and Collisions Reading assignment: Chapter Homework #16 :(due Monday, Oct. 10, 2005): Problems:Q1, Q14, 9, 14, 21, 28

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1) If two objects collide and one is initially at rest, is it possible for both to be at rest after the collision? A) yes B) no 2) Is it possible for one to be at rest after the collision? A) yes B) no

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Elastic and inelastic collisions in one dimension Momentum is conserved in any collision, elastic and inelastic. Kinetic Energy is only conserved in elastic collisions. Perfectly inelastic collision: After colliding, particles stick together. There is a loss of kinetic energy (deformation). Inelastic collisions: Particles bounce off each with some loss of kinetic energy. Perfectly elastic collision: Particles bounce off each other without loss of kinetic energy.

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Perfectly inelastic collision of two particles (Particles stick together) Notice that p and v are vectors and, thus have a direction (+/-) There is a loss in kinetic energy, E loss

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Elastic collision of two particles (Particles bounce off each other without loss of energy. Energy is conserved : Momentum is conserved:

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For elastic collisions in one dimension: Suppose we know the initial masses and velocities. Then: (9.21) (9.22)

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Black board example 9.2 Two carts collide elastically on a frictionless track. The first cart (m 1 = 1kg) has a velocity in the positive x-direction of 2 m/s; the other cart (m = 0.5 kg) has velocity in the negative x-direction of 5 m/s. (a)Find the speed of both carts after the collision. (b)What is the speed if the collision is inelastic? (c)How much energy is lost in the inelastic collision?

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Two-dimensional collisions (Two particles) Conservation of momentum: Split into components: If the collision is elastic, we can also use conservation of energy.

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Black board example 9.3 Accident investigation. Two automobiles of equal mass approach an intersection. One vehicle is traveling towards the east with 29 mi/h (13.0 m/s) and the other is traveling north with unknown speed. The vehicles collide in the intersection and stick together, leaving skid marks at an angle of 55º north of east. The second driver claims he was driving below the speed limit of 35 mi/h (15.6 m/s) m/s ??? m/s Is he telling the truth? What is the speed of the “combined vehicles” right after the collision? How long are the skid marks ( k = 0.5)

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If ball 1 in the arrangement shown here is pulled back and then let go, ball 5 bounces forward. If balls 1 and 2 are pulled back and released, balls 4 and 5 bounce forward, and so on. The number of balls bouncing on each side is equal because 1. of conservation of momentum. 2. the collisions are all elastic. 3. neither of the above

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Impulse (change in momentum) A change in momentum is called “impulse”: During a collision, a force F acts on an object, thus causing a change in momentum of the object: For a constant (average) force: Think of hitting a soccer ball: A force F acting over a time t causes a change p in the momentum (velocity) of the ball.

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A soccer player hits a ball (mass m = 440 g) coming at him with a velocity of 20 m/s. After it was hit, the ball travels in the opposite direction with a velocity of 30 m/s. (a)What impulse acts on the ball while it is in contact with the foot? (b)The impact time is 0.1s. What is the force acting on the ball? Black board example 10.1

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Momentum Momentum is conserved – even in collisions with energy loss due to friction/deformation. Impulse Chapter 9: Linear Momentum and Collisions Reading assignment: Chapter Homework #16 :(due Monday, Oct. 10, 2005): Problems:Q1, Q14, 9, 14, 21, 28

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1) If two objects collide and one is initially at rest, is it possible for both to be at rest after the collision? A) yes B) no 2) Is it possible for one to be at rest after the collision? A) yes B) no

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Elastic and inelastic collisions in one dimension Momentum is conserved in any collision, elastic and inelastic. Kinetic Energy is only conserved in elastic collisions. Perfectly inelastic collision: After colliding, particles stick together. There is a loss of kinetic energy (deformation). Inelastic collisions: Particles bounce off each with some loss of kinetic energy. Perfectly elastic collision: Particles bounce off each other without loss of kinetic energy.

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Perfectly inelastic collision of two particles (Particles stick together) Notice that p and v are vectors and, thus have a direction (+/-) There is a loss in kinetic energy, E loss

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You are given two carts, A and B. They look identical, and you are told that they are made of the same material. You place A at rest on an air track and give B a constant velocity directed to the right so that it collides with A. After the collision, both carts move to the right, the velocity of B being smaller than what it was before the collision. What do you conclude? 1. Cart A is hollow. 2. The two carts are identical. 3. Cart B is hollow. 4. need more information

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A car accelerates from rest. In doing so the car gains a certain amount of momentum and Earth gains 1. more momentum. 2. the same amount of momentum. 3. less momentum. 4. The answer depends on the interaction between the two.

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A car accelerates from rest. It gains a certain amount of kinetic energy and Earth 1. gains more kinetic energy. 2. gains the same amount of kinetic energy. 3. gains less kinetic energy. 4. loses kinetic energy as the car gains it.

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Suppose the entire population of the world gathers in one spot and, at the sounding of a prearranged signal, everyone jumps up. While all the people are in the air, does Earth gain momentum in the opposite direction? 1. No; the inertial mass of Earth is so large that the planet’s change in motion is zero. 2. Yes; because of its much larger inertial mass, however, the change in momentum of Earth is much less than that of all the jumping people. 3. Yes; Earth recoils, like a rifle firing a bullet, with a change in momentum equal to and opposite that of the people. 4. It depends.

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Suppose the entire population of the world gathers in one spot and, at the sound of a prearranged signal, everyone jumps up. About a second later,5 billion people land back on the ground. After the people have landed, Earth’s momentum is 1. the same as what it was before the people jumped. 2. different from what it was before the people jumped.

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Suppose rain falls vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the speed of the cart 1. increases. 2. does not change. 3. decreases.

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Suppose rain falls vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the kinetic energy of the cart 1. increases. 2. does not change. 3. decreases.

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Consider these situations: (i) a ball moving at speed v is brought to rest; (ii) the same ball is projected from rest so that it moves at speed v; (iii) the same ball moving at speed v is brought to rest and then projected backward to its original speed. In which case(s) does the ball undergo the largest change in momentum? 1. (i) 2. (i) and (ii) 3. (ii) 4. (ii) and (iii) 5. (iii)

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Consider two carts, of masses m and 2m, at rest on an air track. If you push first one cart for 3 s and then the other for the same length of time, exerting equal force on each, the momentum of the light cart is 1. four times 2. twice 3. equal to 4. one-half 5. one-quarter the momentum of the heavy cart.

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Consider two carts, of masses m and 2m, at rest on an air track. If you push first one cart for 3 s and then the other for the same length of time, exerting equal force on each, the kinetic energy of the light cart is 1. larger than 2. equal to 3. smaller than the kinetic energy of the heavy car.

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Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare? 1. It takes less time to stop the ping-pong ball. 2. Both take the same time. 3. It takes more time to stop the ping-pong ball.

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Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the distances needed to stop them compare? 1. It takes a shorter distance to stop the ping- pong ball. 2. Both take the same distance. 3. It takes a longer distance to stop the ping- pong ball.

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A cart moving at speed v collides with an identical stationary cart on an airtrack, and the two stick together after the collision. What is their velocity after colliding? 1. v v 3. zero 4. –0.5 v 5. –v 6. need more information

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A person attempts to knock down a large wooden bowling pin by throwing a ball at it. The person has two balls of equal size and mass, one made of rubber and the other of putty. The rubber ball bounces back, while the ball of putty sticks to the pin. Which ball is most likely to topple the bowling pin? 1. the rubber ball 2. the ball of putty 3. makes no difference 4. need more information

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Think fast! You’ve just driven around a curve in a narrow, one-way street at 25 mph when you notice a car identical to yours coming straight toward you at 25 mph. You have only two options: hitting the other car head on or swerving into a massive concrete wall, also head on. In the split second before the impact, you decide to 1. hit the other car. 2. hit the wall. 3. hit either one—it makes no difference. 4. consult your lecture notes.

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If all three collisions in the figure shown here are totally inelastic, which bring(s) the car on the left to a halt? 1. I 2. II 3. III 4. I, II 5. I, III 6. II, III 7. all three

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If all three collisions in the figure shown are totally inelastic, which cause(s) the most damage? 1. I 2. II 3. III 4. I, II 5. I, III 6. II, III 7. all three

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A golf ball is fired at a bowling ball initially at rest and bounces back elastically. Compared to the bowling ball, the golf ball after the collision has 1. more momentum but less kinetic energy. 2. more momentum and more kinetic energy. 3. less momentum and less kinetic energy. 4. less momentum but more kinetic energy. 5. none of the above

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A golf ball is fired at a bowling ball initially at rest and sticks to it. Compared to the bowling ball, the golf ball after the collision has 1. more momentum but less kinetic energy. 2. more momentum and more kinetic energy. 3. less momentum and less kinetic energy. 4. less momentum but more kinetic energy. 5. none of the above

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Suppose you are on a cart, initially at rest on a track with very little friction. You throw balls at a partition that is rigidly mounted on the cart. If the balls bounce straight back as shown in the figure, is the cart put in motion? 1. Yes, it moves to the right. 2. Yes, it moves to the left. 3. No, it remains in place.

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A compact car and a large truck collide head on and stick together. Which undergoes the larger momentum change? 1. car 2. truck 3. The momentum change is the same for both vehicles. 4. Can’t tell without knowing the final velocity of combined mass.

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A compact car and a large truck collide head on and stick together. Which vehicle undergoes the larger acceleration during the collision? 1. car 2. truck 3. Both experience the same acceleration. 4. Can’t tell without knowing the final velocity of combined mass.

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Is it possible for a stationary object that is struck by a moving object to have a larger final momentum than the initial momentum of the incoming object? 1. Yes. 2. No because such an occurrence would violate the law of conservation of momentum.

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Two carts of identical inertial mass are put back- to-back on a track. Cart A has a spring loaded piston; cart B is entirely passive. When the piston is released, it pushes against cart B, and 1. A is put in motion but B remains at rest. 2. both carts are set into motion, with A gaining more speed than B. 3. both carts gain equal speed but in opposite directions. 4. both carts are set into motion, with B gaining more speed than A. 5. B is put in motion but A remains at rest.

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Two carts are put back-to-back on a track. Cart A has a spring-loaded piston; cart B, which has twice the inertial mass of cart A, is entirely passive. When the piston is released, it pushes against cart B, and the carts move apart. How do the magnitudes of the final momenta and kinetic energies compare? 1. pA > pB, kA > kB 2. pA > pB, kA = kB 3. pA > pB, kA < kB 4. pA = pB, kA > kB 5. pA = pB, kA = kB 6. pA = pB, kA < kB 7. pA kB 8. pA < pB, kA = kB 9. pA < pB, kA < kB

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Two carts are put back-to-back on a track. Cart A has a spring-loaded piston; cart B, which has twice the inertial mass of cart A, is entirely passive. When the piston is released, it pushes against cart B, and the carts move apart. Ignoring signs, while the piston is pushing, 1. A has a larger acceleration than B. 2. the two have the same acceleration. 3. B has a larger acceleration than A.

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Two people on roller blades throw a ball back and forth. Which statement(s) is/are true? A. The interaction mediated by the ball is repulsive. B. If we film the action and play the movie backward, the interaction appears attractive. C. The total momentum of the two people is conserved. D. The total energy of the two people is conserved. What about the conservation laws? The ball carries both momentum and energy back and forth between the two roller-bladers. Their momentum and energy therefore cannot be conserved.

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In the following figure, a 10-kg weight is suspended from the ceiling by a spring. The weight-spring system is at equilibrium with the bottom of the weight about 1 m above the floor. The spring is then stretched until the weight is just above the eggs. When the spring is released, the weight is pulled up by the contracting spring and then falls back down under the influence of gravity. On the way down, it 1. reverses its direction of travel well above the eggs. 2. reverses its direction of travel precisely as it reaches the eggs. 3. makes a mess as it crashes into the eggs.

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In part (a) of the figure, an air track cart attached to a spring rests on the track at the position x equilibrium and the spring is relaxed. In (b), the cart is pulled to the position x start and released. It then oscillates about x equilibrium. Which graph correctly represents the potential energy of the spring as a function of the position of the cart?

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Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inertia (Rotational inertia) Torque For every rotational quantity, there is a linear analog. Chapter 10:Rotation of a rigid object about a fixed axis Part 1 Reading assignment: Chapter (know concept of moment of inertia, don’t worry about integral calculation) Homework :(due Wednesday, Oct. 14, 2009): Chapter 10: Q23, 35, 48, 57, 80

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1. The center of mass of a rigid object of arbitrary shape ___ 1. is always inside the object. ___ 2. can lie outside the object. ___ 3. depends on the motion of the object. ___ 4. depends on the frame of reference of the object. 2. Compared with the kinetic energy of its center of mass (CM), the total kinetic energy of a system is ___ 1. always less than the kinetic energy of the CM. ___ 2. always equal to the kinetic energy of the CM. ___ 3. greater than or equal to the kinetic energy of the CM. ___ 4. depends on the particular system 3. A rocket is propelled forward by ejecting gas at high speed. The forward motion is a consequence of ___ 1. conservation of energy. ___ 2. conservation of momentum. ___ 3. both of the above. ___ 4. neither of the above.

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Planar, rigid object rotating about origin O. Rotational motion Look at one point P: Arc length s: Thus, the angular position is: is measured in degrees or radians (more common)

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One radian is the angle subtended by an arc length equal to the radius of the arc. r s = r For full circle: Full circle has an angle of 2 radians, Thus, one radian is 360°/2 Radian degrees 2 360° 180° 90° 157.3°

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Define quantities for circular motion (note analogies to linear motion!!) Angular displacement: Average angular speed: Instantaneous angular speed: Average angular acceleration: Instantaneous angular acceleration:

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A ladybug sits at the outer edge of a merry-go- round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular speed is 1. half the ladybug’s. 2. the same as the ladybug’s. 3. twice the ladybug’s. 4. impossible to determine

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Angular quantities are vectors Angular velocity, angular acceleration, angular momentum, torque. Right-hand rule for determining the direction of this vector. rotates through the same angle, has the same angular velocity, has the same angular acceleration. Every particle (of a rigid object): characterize rotational motion of entire object

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A ladybug sits at the outer edge of a merry-go- round, that is turning and slowing down. At the instant shown in the figure, the radial component of the ladybug’s (Cartesian) acceleration is 1. in the +x direction. 2. in the –x direction. 3. in the +y direction. 4. in the –y direction.

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A ladybug sits at the outer edge of a merry-go- round, that is turning and slowing down. At the instant shown in the figure, the radial component of the ladybug’s (Cartesian) acceleration is 1. in the +x direction. 2. in the –x direction. 3. in the +y direction. 4. in the –y direction.

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A ladybug sits at the outer edge of a merry-go- round that is turning and is slowing down. The vector expressing her angular velocity is 1. in the +x direction. 2. in the –x direction. 3. in the +y direction. 4. in the –y direction. 5. in the +z direction.

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Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration,

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Black board example 11.1 A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s. (a)What is the magnitude of the angular acceleration of the wheel? (b)Through what angle does the wheel rotate in these 3.00 s? (c)Through which angle does the wheel rotate between t = 2.00 s and 3.00 s?

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Relation between linear and angular quantities Tangential speed of a point P: Tangential acceleration of a point P: Note, this is not the centripetal acceleration a r !! Arc length s: Caution: Measure angular quantities in radians

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The diameters of the main rotor and the tail rotor of a helicopter are 7.60 m and 1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4138 rev/min. a)Calculate the speeds of the tips of both rotors. b)Compare with the speed of sound, 343 m/s. Black board example 11.2 c)The rotors are rotating at constant angular speed. What is the centripetal acceleration and what is the angular acceleration?

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Black board example 11.3 HW 27 (a)What is the angular speed about the polar axis of a point on Earth’s surface at a latitude of 40°N (b)What is the linear speed v of that point? (c)What are and v for a point on the equator? Radius of earth: 6370 km

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Two wheels initially at rest roll the same distance without slipping down identical inclined planes starting from rest. Wheel B has twice the radius but the same mass as wheel A. All the mass is concentrated in their rims, so that the rotational inertias are I = mR 2. Which has more translational kinetic energy when it gets to the bottom? 1. Wheel A 2. Wheel B 3. The kinetic energies are the same. 4. need more information

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Consider two people on opposite sides of a rotating merry-go-round. One of them throws a ball toward the other. In which frame of ref- erence is the path of the ball straight when viewed from above: (a) the frame of the merry-go-round or (b) that of Earth? 1. (a) only 2. (a) and (b)—although the paths appear to curve 3. (b) only 4. neither; because it’s thrown while in circular motion, the ball travels along a curved path.

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You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements: a)1, 3, 4, 2 b)2, 4, 1, 3 c)4=2, 1, 3 d)4=2, 1=3 e)2, 1=4, 3

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A force F is applied to a dumbbell for a time interval Δt, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater center-of-mass speed? 1. (a) 2. (b) 3. no difference 4. The answer depends on the rotational inertia of the dumbbell.

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Aforce F is applied to a dumbbell for a time interval Δt, first as in (a) and then as in (b). In which case does the dumbbell acquire the greater energy? 1. (a) 2. (b) 3. no difference 4. The answer depends on the rotational inertia of the dumbbell.

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Imagine hitting a dumbbell with an object coming in at speed v, first at the center, then at one end. Is the center-of-mass speed of the dumbbell the same in both cases? 1. yes 2. no

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A box, with its center-of-mass off-center as indicated by the dot, is placed on an inclined plane. In which of the four orientations shown, if any, does the box tip over?

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Consider the situation shown at left below. A puck of mass m, moving at speed v hits an identical puck which is fastened to a pole using a string of length r. After the collision, the puck attached to the string revolves around the pole. Suppose we now lengthen the string by a factor 2, as shown on the right, and repeat the experiment. Compared to the angular speed in the first situation, the new angular speed is 1. twice as high 3. half as much 2. the same 4. none of the above

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A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational inertia and her angular speed increases so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be 1. the same. 2. larger because she’s rotating faster. 3. smaller since her rotational inertia is smaller.

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Two wheels with fixed hubs, each having a mass of 1 kg, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is I = mR 2. In order to impart identical angular accelerations, how large must F 2 be? N N 3. 1 N 4. 2 N 5. 4 N

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Consider the uniformly rotating object shown below. If the object’s angular velocity is a vector (in other words, it points in a certain direction in space) is there a particular direction we should associate with the angular velocity? 1. yes, ±x 2. yes, ±y 3. yes, ±z 4. yes, some other direction 5. no, the choice is really arbitrary

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A person spins a tennis ball on a string in a horizontal circle (so that the axis of rotation is vertical). At the point indicated below, the ball is given a sharp blow in the forward direction. This causes a change in angular momentum ΔL in the 1. x direction 2. y direction 3. z direction

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A person spins a tennis ball on a string in a horizontal circle (so that the axis of rotation is vertical). At the point indicated below, the ball is given a sharp blow vertically downward. In which direction does the axis of rotation tilt after the blow? 1. +x direction 2. –x direction 3. +y direction 4. –y direction 5. It stays the same (but the magnitude of the angular momentum changes). 6. The ball starts wobbling in all directions.

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A suitcase containing a spinning flywheel is rotated about the vertical axis as shown in (a). As it rotates, the bottom of the suitcase moves out and up, as in (b). From this, we can conclude that the flywheel, as seen from the side of the suitcase as in (a), rotates 1. clockwise. 2. counterclockwise.

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Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inertia (Rotational inertia) Torque For every rotational quantity, there is a linear analog. Chapter 10:Rotation of a rigid object about a fixed axis Part 2 Reading assignment: Chapter Homework :(due Wednesday, Oct. 14, 2009): Problems:Chapter 11 9AE's, 3AF's, Q2, 9, 16, 26, 28

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1. An object is rotated about a vertical axis by 90° and then about a horizontal axis by 180°. If we start over and perform the rotations in the reverse order, the orientation of the object ___ 1. will be the same as before. ___ 2. will be different than before. ___ 3. depends on the shape of the object. 2. A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center of the disk as point P is. The angular velocity of Q at a given time is ___ 1. twice as big as P’s. ___ 2. the same as P’s. ___ 3. half as big as P’s. ___ 4. none of the above. 3. When a disk rotates counterclockwise at a constant rate about a vertical axis through its center, the tangential acceleration of a point on the rim is ___ 1. positive. ___ 2. zero. ___ 3. negative. ___ 4. impossible to determine without more information.

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Rotational energy A rotating object (collection of i points with mass m i ) has a rotational kinetic energy of Where: Rotational inertia

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Demo: Both sticks have the same weight. Why is it so much more difficult to rotate the blue stick?

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Four small spheres are mounted on the corners of a frame as shown. a)What is the rotational energy of the system if it is rotated about the z-axis (out of page) with an angular velocity of 5 rad/s b)What is the rotational energy if the system is rotated about the y- axis? (M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m). Black board example 11.4 What is the rotational inertia?

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Rotational inertia of an object depends on: - the axis about which the object is rotated. - the mass of the object. - the distance between the mass(es) and the axis of rotation.

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Calculation of Rotational inertia for continuous extended objects Refer to Table11-2 Note that the moments of inertia are different for different axes of rotation (even for the same object)

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Rotational inertia for some objects Page 278

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Parallel axis theorem Rotational inertia for a rotation about an axis that is parallel to an axis through the center of mass h What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is rotating about an axis 0.5 away from the center with = 2 rad/sec? Blackboard example 11.4

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Conservation of energy (including rotational energy): Again: If there are no non-conservative forces: Energy is conserved. Rotational kinetic energy must be included in energy considerations!

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Connected cylinders. Two masses m 1 (5 kg) and m 2 (10 kg) are hanging from a pulley of mass M (3 kg) and radius R (0.1 m), as shown. There is no slip between the rope and the pulleys. (a)What will happen when the masses are released? (b)Find the velocity of the masses after they have fallen a distance of 0.5 m. (c)What is the angular velocity of the pulley at that moment? Black board example 11.5

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Torque A force F is acting at an angle on a lever that is rotating around a pivot point. r is the distance between F and the pivot point. This force-lever pair results in a torque on the lever

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Black board example 11.6 Two mechanics are trying to open a rusty screw on a ship with a big ol’ wrench. One pulls at the end of the wrench (r = 1 m) with a force F = 500 N at an angle = 80 °; the other pulls at the middle of wrench with the same force and at an angle = 90 °. What is the net torque the two mechanics are applying to the screw?

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Particle of mass m rotating in a circle with radius r. Radial force F r to keep particle on circular path. Tangential force F t accelerates particle along tangent. Torque and angular acceleration Newton’s 2. law for rotation. Torque acting on particle is proportional to angular acceleration :

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Work in rotational motion: Definition of work: Work in linear motion: Component of force F along displacement s. Angle between F and s. Torque and angular displacement .

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Work and Energy in rotational motion Remember work-kinetic energy theorem for linear motion: There is an equivalent work-rotational kinetic energy theorem: External work done on an object changes its kinetic energy External, rotational work done on an object changes its rotational kinetic energy

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Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration,

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Announcements 1.Midterm 2 on Wednesday, Oct Material: Chapters Review on Tuesday (outside of class time) 4.I’ll post practice tests on Web 5.You are allowed a 3x5 inch cheat card with 10 equations 6.Go through practice exams & homework & class examples; understand concepts & demos 7.Time limit for test: 50 minutes

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1. The rotational inertia of a rigid body ___ 1. is a measure of its resistance to changes in rotational motion. ___ 2. depends on the location of the axis of rotation. ___ 3. is large if most of the body’s mass is far from the axis of rotation. ___ 4. is all of the above. ___ 5. is none of the above. 2. The angular momentum of a particle ___ 1. is independent of the specific origin of coordinates. ___ 2. is zero when its position and momentum vectors are parallel. ___ 3. is zero when its position and momentum vectors are perpendicular. ___ 4. is not covered in the reading assignment. 3. A wheel rolls without slipping along a horizontal surface. The center of the wheel has a translational speed v. The lowermost point on the wheel has a net forward velocity ___ 1. 2v. ___ 2. v. ___ 3. zero. ___ 4. need more information

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Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration,

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Summary: Angular and linear quantities Kinetic Energy: Torque: Linear motion Rotational motion Kinetic Energy: Force: Momentum:Angular Momentum: Work:

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Rolling motion Pure rolling: There is no slipping Linear speed of center of mass:

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Rolling motion The angular velocity of any point on the wheel is the same. The linear speed of any point on the object changes as shown in the diagram!! For one instant (bottom), point P has no linear speed. For one instant (top), point P’ has a linear speed of 2·v CM

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Rolling motion of a particle on a wheel (Superposition of rolling and linear motion) Rolling Rotation Linear + =

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Superposition principle: Rolling motion = Pure translation + Pure rotation Rolling motion Kinetic energy of rolling motion:

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Torque Angular momentum Angular momentum is conserved Chapter 11: Angular Momentum part 1 Reading assignment: Chapter Homework :(due Monday, Oct. 17, 2005): Problems:30, 41, 42, 44, 48, 53

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Torque and the vector product Thus far: Torque Torque is the vector product between the force vector F and vector r

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Torque and the vector product Definition of vector product: - The vector product of vectors A and B is the vector C. - C is perpendicular to A and B - The magnitude of C = A·B·sin

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Torque and the vector product Use the right hand rule to figure out the direction of C. - Thumb is C (i.e. torque angular velocity , angular momentum L ) - Index finger is A (e.g. radius r) - Middle finger is B (e.g. force F)

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Torque and the vector product Rules for the vector product. If A is parallel to B then.Thus, If A is perpendicular to B then Magnitude of C = A·B·sin is equal to area of parallelogram made by A and B

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Torque and the vector product Rules for the vector product (cont). 6.

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A force F = (2.00i j) is applied to an object that is pivoted about a fixed axis aligned along the z-axis. The force is applied at the point r = (4.00i j). Black board example 12.2 (a)What is the torque exerted on the object? (b)What is the magnitude and direction of the torque vector . (c)What is the angle between the directions of F and r?

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Angular momentum of a particle L… angular momentum r… distance from the origin p… momentum of particle v…velocity of particle Definition: L is perpendicular to r and p L has magnitude L = r·p·sin

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Angular momentum of a rotating rigid object We’ll consider an object that is rotating about the z-axis. The angular momentum of the object is given by: Note that in this case L and are along the z axis. Also note the analog formula for linear momentum p = m·v

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Black board example 12.3 A light rigid rod, 1 m in length, joins two particles – with masses 3 kg and 4 kg at its end. The system rotates in the x-y plane about a pivot through the center of the rod. Determine the angular momentum of the system about the origin when the speed of each particle is 5.00 m/s.

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Conservation of angular momentum The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero. If the system undergoes an internal “rearrangement”, then If the object is rotating about a fixed axis (say z-axis), then:

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Conservation laws

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Demo A students stands still on a rotatable platform and holds a spinning wheel. The bicycle wheel is spinning in the clockwise direction when viewed from above. He flips the wheel over. What happens?

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Student on a turn table. A student stands on a platform that is rotating with an angular speed of 7.5 rad/s, his arms outstretched and he holds a brick in each hand. The rotational inertia of the whole system is 6.0 kg·m 2. The student then pulls the bricks inward thus reducing the rotational inertia to 2.0 kg·m 2. (a)What is the new angular speed of the platform? (b)What is the ratio of the new kinetic energy of the system to the original kinetic energy? (c)What provided the added kinetic energy? Black board example 12.4

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Summary: Angular and linear quantities Kinetic Energy: Torque: Linear motion Rotational motion Kinetic Energy: Force: Momentum:Angular Momentum: Work:

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