1. Rotational Kinematics and Energy (Cont.)
Chapter 10
Rotational Kinematics and Energy (Cont.)
Dr. Jie Zou PHY 1151G Department of Physics

2. Dr. Jie Zou PHY 1151G Department of Physics
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

3. Dr. Jie Zou PHY 1151G Department of Physics
(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

4. (2) Centripetal and tangential acceleration
Centripetal acceleration acp:
The acceleration due to changing direction of motion.
Magnitude: acp = v2/r = r2; 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
Dr. Jie Zou PHY 1151G Department of Physics

5. 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

6. Rolling motion
The linear (or translational) speed of the axle of a rolling object: v = 2r /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)

7. Dr. Jie Zou PHY 1151G Department of Physics
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 = m/s).
Answers: (a) 77 rad/s. (b) 110 mph.
Dr. Jie Zou PHY 1151G Department of Physics

8. Rotational kinetic energy and the moment of inertia
SI unit: J
I: Moment of inertia (SI unit: kg·m2)
: Angular speed (SI unit: rad/s)
Moment of inertia, I
I =  miri2
SI unit: kg·m2
Moment of inertia, I, depends on:
Distribution of mass
Location and orientation of the axis of rotation
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

9. 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)
Dr. Jie Zou PHY 1151G Department of Physics

10. Table 10-1 Moments of inertia for objects of various shapes
Dr. Jie Zou PHY 1151G Department of Physics

11. Dr. Jie Zou PHY 1151G Department of Physics
Real world examples
Dr. Jie Zou PHY 1151G Department of Physics

12. 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?

13. Dr. Jie Zou PHY 1151G Department of Physics
Homework
See online homework assignment on
Dr. Jie Zou PHY 1151G Department of Physics