Dr. Jie Zou PHY 1151G Department of Physics

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Chapter 10

Rotational Kinematics and Energy (Cont.)

Dr. Jie Zou PHY 1151G Department of Physics

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Outline Connections between linear and

rotational quantities (1) Tangential speed (2) Centripetal acceleration and tangential

acceleration Real world examples (BIO) Rolling motion Rotational kinetic energy and the

moment of inertia Conservation of energy

Dr. Jie Zou PHY 1151G Department of Physics

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(1) Tangential speed Tangential speed v: The speed

of something moving along a circular path; the direction of motion is always tangential to the circle; SI units: m/s.

Relation between tangential and angular speed: v = r . : The angular speed; it must be in

rad/s. is the same for every point on a

rotating object; v is greater on the outside of a rotating object than inside and closer to the axis.

Dr. Jie Zou PHY 1151G Department of Physics

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(2) Centripetal and tangential acceleration

Centripetal acceleration acp: The acceleration due to changing

direction of motion. Magnitude: acp = v2/r = r2; SI unit:

m/s2

Direction: Directed towards the axis of rotation

Tangential acceleration at: The acceleration due to changing

angular speed. Magnitude: at = r; SI unit: m/s2

Direction: Tangential to the circular path

Microhematocrit centrifuge

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Real world example (BIO) Example 10-3 The

microhematocrit: In a microhematocrit centrifuge, small samples of blood are placed in heparinized capillary tubes (heparin is an anticoagulant). The tubes are rotated at 11,500 rpm, with the bottom of the tubes 9.07 cm from the axis of rotation.

(a) Find the linear speed of the bottom of the tubes.

(b) What is the centripetal acceleration at the bottom of the tubes?

Dr. Jie Zou PHY 1151G Department of Physics

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Rolling motion The linear (or translational) speed of the axle

of a rolling object: v = 2r /T = r = vt

A rolling object combines rotational motion and translation motion: (a) pure rotational motion; (b) pure translational motion; (c) rolling without slipping

(a) (b) (c)

Dr. Jie Zou PHY 1151G Department of Physics

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Example 10-6 A car with tires of radius

32 cm drives on the highway at 55 mph. (a) What is the angular speed of the tires? (b) What is the linear speed of the top of the tires (1 mph = 0.447 m/s). Answers: (a) 77 rad/s. (b)

110 mph.

Dr. Jie Zou PHY 1151G Department of Physics

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Rotational kinetic energy and the moment of inertia

Rotational kinetic energy

SI unit: J I: Moment of inertia (SI unit: kg·m2) : Angular speed (SI unit: rad/s)

Moment of inertia, I I = miri

2 SI unit: kg·m2

Moment of inertia, I, depends on:

Distribution of mass Location and orientation of the axis

of rotation

K 1

2I2

The moment of inertia plays the same role in rotational motion that mass plays in translational motion.

Dr. Jie Zou PHY 1151G Department of Physics

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Examples: moment of inertia (a) Use the general definition,

find the moment of inertia, I, for the dumbbell-shaped object.

(b) If the two masses are moved closer to the axis of rotation, such that each has a radius of R/2, what is I now?

(c) If the object is rotated about one end, what is its moment of inertia, I, about this new axis of rotation?

(a)

(c)

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Table 10-1 Moments of inertia for objects of various shapes

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Real world examples

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Application of conservation of mechanical energy to a rolling object

Kinetic energy of rolling motion:

The kinetic energy of a rolling object is the sum of its translational kinetic energy and its rotational kinetic energy.

Example: An object of mass m, radius r, and moment of inertia I at the top of a ramp is released from rest and rolls without slipping to the bottom, a vertical height h below the starting point. What is the object’s speed on reaching the bottom?

K 1

2mv2

1

2I 2

Dr. Jie Zou PHY 1151G Department of Physics

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Homework

See online homework assignment on www.masteringphysics.com