**Chapter 7** Rotational Motion • Angular and tangential acceleration • Linear and rotational motion compared • Torque • Center of gravity • Newton’s second law for rotation Topics: Sample question: As the earth rotates on its axis,the distant stars appear to move in eternal circles in the sky overhead. In reality, however, the angular velocity of the earth is very slowly decreasing, leading to an increase in the length of the day of 18 μs each year. What causes the angular velocity of a rotating object to change? Slide 7-1

**Reading Quiz** • Moment of inertia is • the rotational equivalent of mass. • the point at which all forces appear to act. • the time at which inertia occurs. • an alternative term for moment arm. Slide 7-2

**Answer** • Moment of inertia is • the rotational equivalent of mass. Slide 7-3

**Reading Quiz** • Which factor does the torque on an object not depend on? • The magnitude of the applied force. • The object’s angular velocity. • The angle at which the force is applied. • The distance from the axis to the point at which the force is applied. Slide 7-4

**Answer** • Which factor does the torque on an object not depend on? • The object’s angular velocity. Slide 7-5

**Reading Quiz** • Which statement about an object’s center of gravity is not true? • If an object is free to rotate about a pivot, the center of gravity will come to rest below the pivot. • The center of gravity coincides with the geometric center of the object. • The torque due to gravity can be calculated by considering the object’s weight as acting at the center of gravity. • For objects small compared to the earth, the center of gravity and the center of mass are essentially the same. Slide 7-6

**Answer** • Which statement about an object’s center of gravity is not true? • The center of gravity coincides with the geometric center of the object. Slide 7-7

**Reading Quiz** • A net torque applied to an object causes • a linear acceleration of the object. • the object to rotate at a constant rate. • the angular velocity of the object to change. • the moment of inertia of the object to change. Slide 7-8

**Answer** • A net torque applied to an object causes • the angular velocity of the object to change. Slide 7-9

**Angular Acceleration** Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-10

**Linear and Circular Motion Compared** Slide 7-11

**Linear and Circular Kinematics Compared** Slide 7-12

**Example** A high-speed drill rotating CCW takes 2.5 s to speed up to 2400 rpm. What is the drill’s angular acceleration? B. How many revolutions does it make as it reaches top speed? Slide 7-13

**Tangential Acceleration** Slide 7-14

**Torque** Which force would be most effective in opening the door? Slide 7-15

**Interpreting Torque** Torque is due to the component of the force perpendicular to the radial line. Slide 7-16

**A Second Interpretation of Torque** Slide 7-17

**Example** Revolutionaries attempt to pull down a statue of the Great Leader by pulling on a rope tied to the top of his head. The statue is 17 m tall, and they pull with a force of 4200 N at an angle of 65° to the horizontal. What is the torque they exert on the statue? If they are standing to the right of the statue, is the torque positive or negative? Slide 7-18

**Center of Gravity** = Slide 7-19

**Calculating the Center-of-Gravity Position** Slide 7-20

**Example** An object consists of the three balls shown, connected by massless rods. Find the x- and y-positions of the object’s center of gravity. Slide 7-21

**Checking Understanding** Which point could be the center of gravity of this L-shaped piece? Slide 7-22

**Answer** Which point could be the center of gravity of this L-shaped piece? Slide 7-23

**Newton’s Second Law for Rotation** I = moment of inertia. Objects with larger moments of inertia are harder to get rotating. Slide 7-24

**Moments of Inertia of Common Shapes** Slide 7-25

**Rotational and Linear Dynamics Compared** Slide 7-26

**Slide 7-27**

**Example** The motor in a CD player exerts a torque of 7.0 x 10-4 N · m. What is the disk’s angular acceleration? (A CD has a diameter of 12.0 cm and a mass of 16 g.) Slide 7-28

**Example** A baseball bat has a mass of 0.82 kg and is 0.86 m long. It’s held vertically and then allowed to fall. What is the bat’s angular acceleration when it has reached 20° from the vertical? (Model the bat as a uniform cylinder). Slide 7-29

**Constraints Due to Ropes and Pulleys** Slide 7-30

**Example** How long does it take the small mass to fall 1.0 m when released from rest? Slide 7-31