Circular Motion, Gravitation, Rotation, Bodies in Equilibrium Chapter 6,9,10 Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

Circular Motion Ball at the end of a string revolving Planets around Sun Moon around Earth

The Radian The radian is a unit of angular measure The radian can be defined as the arc length s along a circle divided by the radius r 57.3°

More About Radians Comparing degrees and radians Converting from degrees to radians

Angular Displacement Axis of rotation is the center of the disk Need a fixed reference line During time t, the reference line moves through angle θ

Angular Displacement, cont. The angular displacement is defined as the angle the object rotates through during some time interval The unit of angular displacement is the radian Each point on the object undergoes the same angular displacement

Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

Angular Speed, cont. The instantaneous angular speed Units of angular speed are radians/sec rad/s Speed will be positive if θ is increasing (counterclockwise) Speed will be negative if θ is decreasing (clockwise)

Average Angular Acceleration The average angular acceleration of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

Angular Acceleration, cont Units of angular acceleration are rad/s² Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

Angular Acceleration, final The sign of the acceleration does not have to be the same as the sign of the angular speed The instantaneous angular acceleration

Analogies Between Linear and Rotational Motion Linear Motion with constant acc. (x,v,a) Rotational Motion with fixed axis and constant a (q,,a)

Examples 78 rev/min=? A fan turns at a rate of 900 rpm Tangential speed of tips of 20cm long blades? Now the fan is uniformly accelerated to 1200 rpm in 20 s

Relationship Between Angular and Linear Quantities Displacements Speeds Accelerations Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion

Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration The centripetal acceleration is due to the change in the direction of the velocity

Centripetal Acceleration, cont. Centripetal refers to “center-seeking” The direction of the velocity changes The acceleration is directed toward the center of the circle of motion

Centripetal Acceleration, final The magnitude of the centripetal acceleration is given by This direction is toward the center of the circle

Centripetal Acceleration and Angular Velocity The angular velocity and the linear velocity are related (v = ωR) The centripetal acceleration can also be related to the angular velocity

Forces Causing Centripetal Acceleration Newton’s Second Law says that the centripetal acceleration is accompanied by a force F = ma  F stands for any force that keeps an object following a circular path Tension in a string Gravity Force of friction

Examples Ball at the end of revolving string Fast car rounding a curve

More on circular Motion Length of circumference = 2R Period T (time for one complete circle)

Example 200 grams mass revolving in uniform circular motion on an horizontal frictionless surface at 2 revolutions/s. What is the force on the mass by the string (R=20cm)?

Newton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

Universal Gravitation, 2 G is the constant of universal gravitational G = 6.673 x 10-11 N m² /kg² This is an example of an inverse square law

Universal Gravitation, 3 The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1 The forces form a Newton’s third law action-reaction

Universal Gravitation, 4 The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center

Gravitation Constant Determined experimentally Henry Cavendish 1798 The light beam and mirror serve to amplify the motion

Applications of Universal Gravitation Weighing the Earth

Applications of Universal Gravitation “g” will vary with altitude

Escape Speed The escape speed is the speed needed for an object to soar off into space and not return For the earth, vesc is about 11.2 km/s Note, v is independent of the mass of the object

Various Escape Speeds The escape speeds for various members of the solar system Escape speed is one factor that determines a planet’s atmosphere

Motion of Satellites Consider only circular orbit Radius of orbit r: Gravitational force is the centripetal force.

Motion of Satellites Period  Kepler’s 3rd Law

Communications Satellite A geosynchronous orbit Remains above the same place on the earth The period of the satellite will be 24 hr r = h + RE Still independent of the mass of the satellite

Satellites and Weightlessness weighting an object in an elevator Elevator at rest: mg Elevator accelerates upward: m(g+a) Elevator accelerates downward: m(g+a) with a<0 Satellite: a=-g!!

Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related

Torque The door is free to rotate about an axis through O There are three factors that determine the effectiveness of the force in opening the door: The magnitude of the force The position of the application of the force The angle at which the force is applied

Torque, cont Torque, t, is the tendency of a force to rotate an object about some axis t is the torque F is the force symbol is the Greek tau l is the length of lever arm SI unit is N.m Work done by torque W=

Direction of Torque If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

Multiple Torques When two or more torques are acting on an object, the torques are added If the net torque is zero, the object’s rate of rotation doesn’t change

Torque and Equilibrium First Condition of Equilibrium The net external force must be zero This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium This is a statement of translational equilibrium

Torque and Equilibrium, cont To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational The Second Condition of Equilibrium states The net external torque must be zero

Equilibrium Example The woman, mass m, sits on the left end of the see-saw The man, mass M, sits where the see-saw will be balanced Apply the Second Condition of Equilibrium and solve for the unknown distance, x

Moment of Inertia The angular acceleration is inversely proportional to the analogy of the mass in a rotating system This mass analog is called the moment of inertia, I, of the object SI units are kg m2

Newton’s Second Law for a Rotating Object The angular acceleration is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object

More About Moment of Inertia There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation

Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m1 … These segments are equidistant from the axis

Other Moments of Inertia

Example Wheel of radius R=20 cm and I=30kg·m². Force F=40N acts along the edge of the wheel. Angular acceleration? Angular speed 4s after starting from rest? Number of revolutions for the 4s? Work done on the wheel?

Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy KEr=½Iω2 Energy concepts can be useful for simplifying the analysis of rotational motion Units (rad/s)!!

Total Energy of a System Conservation of Mechanical Energy Remember, this is for conservative forces, no dissipative forces such as friction can be present Potential energies of any other conservative forces could be added

Rolling down incline Energy conservation Linear velocity and angular speed are related v=R Smaller I, bigger v, faster!!

Work-Energy in a Rotating System In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy (KEt+KER+PE)i+W=(KEt+KER+PE)f

Angular Momentum Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Angular momentum is defined as L = I ω and

Angular Momentum, cont If the net torque is zero, the angular momentum remains constant Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. That is, when

Conservation Rules, Summary In an isolated system, the following quantities are conserved: Mechanical energy Linear momentum Angular momentum

Conservation of Angular Momentum, Example With hands and feet drawn closer to the body, the skater’s angular speed increases L is conserved, I decreases, w increases

Example A 500 grams uniform sphere of 7.0 cm radius spins at 30 rev/s on an axis through its center. Moment of inertia Rotational kinetic energy Angular momentum

Example Find work done to open 30 a 1m wide door with a steady force of 0.9N at right angle to the surface of the door.

Example A turntable is a uniform disk of metal of mass 1.5 kg and radius 13 cm. What torque is required to drive the turntable so that it accelerates at a constant rate from 0 to 33.3 rpm in 2 seconds?

Center of Gravity The force of gravity acting on an object must be considered In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point

Calculating the Center of Gravity The object is divided up into a large number of very small particles of weight (mg) Each particle will have a set of coordinates indicating its location (x,y)

Calculating the Center of Gravity, cont. We wish to locate the point of application of the single force whose magnitude is equal to the weight of the object, and whose effect on the rotation is the same as all the individual particles. This point is called the center of gravity of the object

Coordinates of the Center of Gravity The coordinates of the center of gravity can be found

Center of Gravity of a Uniform Object The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry. Often, the center of gravity of such an object is the geometric center of the object.

Example Find the center of mass (gravity) of these masses: 3kg (0,1), 2kg (0,0) And 1kg (2,0)

Example Find the center of mass (gravity) of the dumbbell, 4 kg and 2 kg with a 4m long 3kg rod.

Torque, review SI unit is N.m t is the torque F is the force symbol is the Greek tau l is the length of lever arm SI unit is N.m

Direction of Torque If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

Multiple Torques When two or more torques are acting on an object, the torques are added If the net torque is zero, the object’s rate of rotation doesn’t change

Example A 2 m by 2 m square metal plate rotates about its center. Calculate the torque of all five forces each with magnitude 50N.

Torque and Equilibrium First Condition of Equilibrium The net external force must be zero The Second Condition of Equilibrium states The net external torque must be zero

Example The system is in equilibrium. Calculate W and find the tension in the rope (T).

Example A 160 N boy stands on a 600 N concrete beam in equilibrium with two end supports. If he stands one quarter the length from one support, what are the forces exerted on the beam by the two supports?