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**Rotational Kinematics**

Lecture 14 Rotational Kinematics

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Reading and Review

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**Rotational Motion Arc length s, measured in radians: s = rθ**

Analogies between linear and rotational kinematics:

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**Linear and Angular Velocity**

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Bonnie and Klyde Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Klyde and Bonnie have about the same mass. Who has the larger kinetic energy? a) Klyde b) Bonnie c) both the same d) the net kinetic energy for both of them is zero since there is no average motion of center of mass Answer: b Bonnie Klyde

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Bonnie and Klyde Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Klyde and Bonnie have about the same mass. Who has the larger kinetic energy? a) Klyde b) Bonnie c) both the same d) the net kinetic energy for both of them is zero since there is no average motion of center of mass Answer: b Their linear speeds v will be different because v = r and Bonnie is located farther out (larger radius r) than Klyde. Bonnie Klyde Therefore KEBonnie > KEKlyde

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Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion: + =

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Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

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Truck Speedometer Suppose that the speedometer of a truck is set to read the linear speed of the truck but uses a device that actually measures the angular speed of the tires. If larger diameter tires are mounted on the truck instead, how will that affect the speedometer reading as compared to the true linear speed of the truck? a) speedometer reads a higher speed than the true linear speed b) speedometer reads a lower speed than the true linear speed c) speedometer still reads the true linear speed

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Truck Speedometer Suppose that the speedometer of a truck is set to read the linear speed of the truck but uses a device that actually measures the angular speed of the tires. If larger diameter tires are mounted on the truck instead, how will that affect the speedometer reading as compared to the true linear speed of the truck? a) speedometer reads a higher speed than the true linear speed b) speedometer reads a lower speed than the true linear speed c) speedometer still reads the true linear speed The linear speed is v = R. So when the speedometer measures the same angular speed as before, the linear speed v is actually higher, because the tire radius is larger than before.

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**Jeff of the Jungle swings on a vine that is 7. 20 m long**

Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time. (b) What centripetal acceleration does Jeff experience at the bottom of his swing? (c) What exerts the force that is responsible for Jeff’s centripetal acceleration?

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**(a) Find Jeff’s angular speed at this time. **

Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time. (b) What centripetal acceleration does Jeff experience at the bottom of his swing? (c) What exerts the force that is responsible for Jeff’s centripetal acceleration? a) b) c) This is the force that is responsible for keeping Jeff in circular motion: the vine.

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**Rotational Kinetic Energy**

For this mass,

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**Rotational Kinetic Energy**

For these two masses, Ktotal = K1 + K2 = mr2ω2

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**Moment of Inertia We can also write the kinetic energy as**

Where I, the moment of inertia, is given by What is the moment of inertia for two equal masses on the ends of a (massless) rod, spinning about the center of the rod?

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**Moment of Inertia We can also write the kinetic energy as**

Where I, the moment of inertia, is given by What is the moment of inertia for a uniform ring of mass M and radius R, rolling around the center of the ring?

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**Moment of Inertia for various shapes**

Moments of inertia of various objects can be calculated: and so one can calculate the kinetic energy for rotational motion:

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A block of mass 1.5 kg is attached to a string that is wrapped around the circumference of a wheel of radius 30 cm and mass 5.0 kg, with uniform mass density. Initially the mass and wheel are at rest, but then the mass is allowed to fall. What is the velocity of the mass after it falls 1 meter?

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A block of mass 1.5 kg is attached to a string that is wrapped around the circumference of a wheel of radius 30 cm and mass 5.0 kg, with uniform mass density. Initially the mass and wheel are at rest, but then the mass is allowed to fall. What is the velocity of the mass after it falls 1 meter?

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**Energy of a rolling object**

The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: Since velocity and angular velocity are related for rolling objects, the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.

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**Conservation of Energy**

An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom?

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**Conservation of Energy**

An object rolls down a ramp - what is its translation and rotational kinetic energy at the bottom? From conservation of energy: Note: no dependence on mass, only on distribution of mass Velocity at any height:

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**Conservation of Energy**

If these two objects, of the same radius, are released simultaneously, which will reach the bottom first? The disk will reach the bottom first – it has a smaller moment of inertia. More of its gravitational potential energy becomes translational kinetic energy, and less rotational.

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**ConcepTest at a greater height as when it was released.**

A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is: at a greater height as when it was released. at a lesser height as when it was released. at the same height as when it was released. impossible to tell without knowing the mass of the ball. impossible to tell without knowing the radius of the ball.

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**Q: What if both sides of the half-pipe were no-slip?**

ConcepTest A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in the figure. When the ball reaches its maximum height on the frictionless surface, it is: at a greater height as when it was released. at a lesser height as when it was released. at the same height as when it was released. impossible to tell without knowing the mass of the ball. impossible to tell without knowing the radius of the ball. Q: What if both sides of the half-pipe were no-slip?

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The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be increasing by x 10-5 s per year. What is the angular acceleration of the pulsar in rad/s2 ? Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? Under the same assumption, what was the period of the pulsar when it was created?

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The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be decreasing by x 10-5 s per year. What is the angular acceleration of the pulsar in rad/s2 ? Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? Under the same assumption, what was the period of the pulsar when it was created?

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The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be decreasing by x 10-5 s per year. What is the angular acceleration of the pulsar in rad/s2 ? Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? Under the same assumption, what was the period of the pulsar when it was created? (a)

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The pulsar in the Crab nebula was created by a supernova explosion that was observed on Earth in a.d Its current period of rotation (33.0 ms) is observed to be decreasing by x 10-5 s per year. What is the angular acceleration of the pulsar in rad/s2 ? Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? Under the same assumption, what was the period of the pulsar when it was created? (b) (c)

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**Power output of the Crab pulsar**

Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ? The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star. calculate the rotational kinetic energy at the beginning and at the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

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**Power output of the Crab pulsar**

Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W (which is about 150,000 times the power output of our sun). Since the pulsar is out of nuclear fuel, where does all this energy come from ? The angular speed of the pulsar, and so the rotational kinetic energy, is going down over time. This kinetic energy is converted into the energy coming out of that star. calculate the rotational kinetic energy at the beginning and at the end of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and the initial angular speed to be 190 s-1. Δω over one second is given by the angular acceleration.

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Rotational Dynamics

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Torque We know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long- handled wrenches, and why doorknobs are not next to hinges.

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**The torque increases as the force increases, and also as the distance increases.**

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**Only the tangential component of force causes a torque**

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**A more general definition of torque:**

Fsinθ Fcosθ Right Hand Rule You can think of this as either: - the projection of force on to the tangential direction OR - the perpendicular distance from the axis of rotation to line of the force

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Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.

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**e) all are equally effective**

Using a Wrench a c d b You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in tightening the nut? Answer: b e) all are equally effective

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**e) all are equally effective**

Using a Wrench a c d b You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in tightening the nut? Because the forces are all the same, the only difference is the lever arm. The arrangement with the largest lever arm (case #2) will provide the largest torque. e) all are equally effective

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**The gardening tool shown is used to pull weeds. If a 1**

The gardening tool shown is used to pull weeds. If a 1.23 N-m torque is required to pull a given weed, what force did the weed exert on the tool? What force was used on the tool?

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**Force and Angular Acceleration**

Consider a mass m rotating around an axis a distance r away. Newton’s second law: a = r α Or equivalently,

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**Torque and Angular Acceleration**

Once again, we have analogies between linear and angular motion:

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The L-shaped object shown below consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.2 rad/s2 if it is rotated about (a) the x axis, (b) the y axis (c) the z axis (through the origin and perpendicular to the page) (a) (b) (c)

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**Provided in lecture notes on: 10/19**

PHYS2010 midterm 2, Fall 2008 4) Two uniform solid spheres have the same mass, but one has twice the radius of the other. The ratio of the larger sphere's moment of inertia to that of the smaller sphere is A) 8/5 B) 4 C) 2 D) 1/2 E) 4/5 suggested time: 1-2 minutes Please do not ask questions about this problem at discussion sessions before 10/21

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**Provided in lecture notes on: 10/19**

PHYS2010 midterm 2, Fall 2008 7) A solid disk is released from rest and rolls without slipping down an inclined plane that makes an angle of 25.0o with the horizontal. What is the speed of the disk after it has rolled 3.00 m, measured along the plane? (Moment of inertia of a solid disk of mass M and radius R is 1/2(MR2)). A) 5.71 m/s B) 2.04 m/s C) 3.53 m/s D) 4.07 m/s E) 6.29 m/s suggested time: 4-5 minutes Please do not ask questions about this problem at discussion sessions before 10/21

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