Rotation of a Rigid Body
Rigid Body – is an extended object whose size and shape do not change as it moves.
Types of motion of a rigid body
Every point on a rotating rigid body has the same angular velocity
Angular Acceleration – is the rate of change of angular velocity over time.
Example 1
A high-speed drill rotating CCW takes 2.5 s to speed up to 2400
rpm.
(a) What is the drill’s angular acceleration?
(b) How many revolutions does it make as it reaches top speed?
Tangential Acceleration
Torque
Torque is the rotational equivalent of force.
Torque
A torque that tends to rotate the object in a counterclockwise direction is positive, while a negative torque gives a clockwise rotation.
Example 2
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.
(a) What is the torque they exert on the statue?
(b) If they are standing to the right of the statue, is the torque positive or negative?
Gravitational Torque and the Center of Gravity
Finding the Center of Gravity
Example 3
A 1.7 m long barbell has a 20 kg weight on its left end and a 35 kg weight on its right end.
(a) If you ignore the weight of the bar itself, how far from the left end of the barbell is the center of gravity?
(b) Where is the center of gravity if the 8.0 kg mass of the barbell itself is taken into account?
Rotational Dynamics and Moment of Inertia
A torque causes an angular acceleration.
Moment of Inertia – For a system of point masses, the sum of the product of each mass and the square of its perpendicular (minimum) distance from a given axis.
Newton's Second Law for Rotation
Moment of Inertia is the rotational equivalent of mass
Example 4
The figure shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk. Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle.
Rolling Motion
Example 5
A 1.2 g pebble is stuck in a tread of a 0.75 m diameter automobile tire, held in place by a static friction that can be at most 3.6 N. The car starts from rest and gradually accelerates on a straight road. How fast is the car moving when the pebble flies out of the tire tread?