Chapter 11 Examples

Example 1 (HRW)

The angular position of a reference line on a spinning wheel is given by

q(t) = t³ - 27t + 4

(a) Find w(t) and a(t).

(b) Find q(0), w(0) and a(0).

(c) Describe the motion for t > 0.

Key concepts: w and a are first and second derivatives of the angular displacement q .

Example 2 (HRW)

A grindstone is initially rotating at w_o = -4.6 rad/sec.  It is undergoing a constant angular accelertion a = 0.35 rad/sec²

(a) At what time will it momentarily stop?

(b) What time corresponds to Dq = 5 rev.

Key concepts: Use of the equations relating w, a and q for the case of constant a.

w_f = w_o + at            Dq = w_ot + (1/2)at²

Example 3 (HRW)

A uniform disk of mass M = 2.5 kg and radius = 0.2 m is mounted on a fixed horizontal axle.  A block of mass 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk.  Find the acceleration of the falling block, angular acceleration of the disk, and the tension in the cord.  The cord does not slip and there is no friction in the axle.

Key concepts: Newtons second law for rotation (massive pulley), sign convention for linear and rotational motion.

St = Ia

Example 4 (HRW)

A rigid sculpture, consists of a thin hoop (of mass m and radius R = 0.15 m) and two thin rods (each of mass m and length = 2.0R).  The rods formed an inverted T and the hoop is the base.   The sculpture can pivot around a hoizontal axis in the plane of the hoop, passing through its center.

a) Find the  moment of inertia I about the specified axis in terms of m and R.

b) Starting from rest, the sculpture rotates around the rotation axis from the initial upright orientation.  What is its angular speed about the axis when it is inverted ?

Key concepts: For part (a) calculate I for the 2 rods and the hoop separately about the specified axis (use PAT) and then add them (ans. = 5.8mR²).   For part (b), follow the atrategy of replacing the sculpture with a center of mass (CM) picture.  Unlike the class problem, this is not trivial.  You will have to find the CM.  Then just use conservation of energy.