Copyright © 2009 Pearson Education, Inc. Chapter 10 Rotational Motion

Question Two objects are sitting on a horizontal table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true? A) Objects 1 and 2 have the same linear velocity, v, and the same angular velocity, . B) Objects 1 and 2 have the same linear velocity, v, and the different angular velocities, . C) Objects 1 and 2 have different linear velocities, v, and the same angular velocity, . D) Objects 1 and 2 have different linear velocities, v, and the different angular velocities, . 1 2

10-1 Angular Quantities Example 10-3: Angular and linear velocities and accelerations. A carousel is initially at rest. At t = 0 it is given a constant angular acceleration α = rad/s 2, which increases its angular velocity for 8.0 s. At t = 8.0 s, determine the magnitude of the following quantities: (a) the angular velocity of the carousel; (b) the linear velocity of a child located 2.5 m from the center; (c) the tangential (linear) acceleration of that child; (d) the centripetal acceleration of the child; and (e) the total linear acceleration of the child.

10-1 Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is R. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: where l is the arc length.

Angular displacement: The average angular velocity is defined as the total angular displacement divided by time: The instantaneous angular velocity: 10-1 Angular Quantities

The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration: 10-1 Angular Quantities

Every point on a rotating body has an angular velocity ω and a linear velocity v. They are related: 10-1 Angular Quantities

If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

10-1 Angular Quantities Here is the correspondence between linear and rotational quantities:

10-1 Angular Quantities Example 10-4: Hard drive. The platter of the hard drive of a computer rotates at 7200 rpm (rpm = revolutions per minute = rev/min). (a) What is the angular velocity (rad/s) of the platter? (b) If the reading head of the drive is located 3.00 cm from the rotation axis, what is the linear speed of the point on the platter just below it? (c) If a single bit requires 0.50 μm of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?

10-1 Angular Quantities The frequency is the number of complete revolutions per second: Frequencies are measured in hertz: The period is the time one revolution takes:

10-2 Vector Nature of Angular Quantities The angular velocity vector points along the axis of rotation, with the direction given by the right-hand rule. If the direction of the rotation axis does not change, the angular acceleration vector points along it as well.

10-3 Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

10-3 Constant Angular Acceleration Example 10-6: Centrifuge acceleration. A centrifuge rotor is accelerated from rest to 20,000 rpm in 30 s. (a) What is its average angular acceleration? (b) Through how many revolutions has the centrifuge rotor turned during its acceleration period, assuming constant angular acceleration?

27.(II) Two blocks, each of mass m, are attached to the ends of a massless rod which pivots as shown in Fig. 10–48. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system when it is first released Torque : Problem 27