(a) Show that the rotational inertia of a solid cylinder of mass M and radius R about its central axis is equal to the rotational inertia of a thin hoop of mass M and radius about its central axis. (b) Show that the rotational inertia I of any given body of mass M about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass M and a radius k given by $\text{k}=\sqrt{\frac{\text{I}}{\text{M}}}$The radius k of the equivalent hoop is called the radius of gyration of the given body.

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length $\mathbf{55}\mathbf{.0}\mathbf{}\mathbf{\text{\hspace{0.17em}m}}$. At the instant it makes an angle of $\mathbf{\text{35.0°}}$with the vertical as it falls, what are

(a) the radial acceleration of the top, and

(b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.)

(c) At what angle $\mathbf{\text{θ}}$ is the tangential acceleration equal to g?

What is the angular speed of (a) the second hand, (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of $\mathbf{T}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.033}\mathbf{\text{\hspace{0.17em}s}}\mathbf{}$ that is increasing at the rate of $\mathbf{1.26}\mathbf{}\mathbf{\times }\mathbf{}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{\text{\hspace{0.17em}s}}\mathbf{/}\mathbf{\text{y}}$ .

(a) What is the pulsar’s angular acceleration $\mathbf{\text{α}}$?

(b) If $\mathbf{\text{α}}$ is constant, how many years from now will the pulsar stop rotating?

(c) The pulsar originated in a supernova explosion seen in the year $\mathbf{1054}$ .Assuming constant a, find the initial T.

An automobile crankshaft transfers energy from the engine to the axle at the rate of $\mathbf{100}\mathbf{\text{\hspace{0.17em}hp}}\mathbf{}\mathbf{(}\mathbf{=}\mathbf{}\mathbf{74}\mathbf{.6}\mathbf{}\mathbf{\text{\hspace{0.17em}kW}}\mathbf{)}$ when rotating at a speed of $\mathbf{1800}\mathbf{\text{\hspace{0.17em}rev}}\mathbf{/}\mathbf{\text{min}}$. What torque (in newton-meters) does the crankshaft deliver?

A record turntable rotating at $\mathbf{33}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{}\mathbf{}\mathbf{\text{rev}}\mathbf{/}\mathbf{\text{min}}$slows down and stops in $\mathbf{30}\mathbf{}\mathbf{}\mathbf{\text{s}}\mathbf{}$after the motor is turned off.

(a) Find its (constant) angular acceleration in revolutions per minute-squared.

(b) How many revolutions does it make in this time?