# Acharya Narendra Dev College/Learning Physics through Problem Solving/Mechanics

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## Problem 1

How do you determine Rotational Inertia of Geometrical Bodies?

## Problem 2

A wheel A of radius $r_a$ = 10.0 cm is coupled by belt B to wheel C of radius $r_c$ = 25.0 cm. Wheel A increases its angular speed from rest at a uniform rate of 1.6 rad/$s^2$. Find the time that wheel C will take to reach a rotational speed of 100 rev/min, assuming that belt does not slip.

## Problem 3

A pulsar is a rapidly rotating neutron star that emits radio pulses with precise synchronisation, one such pulse for each rotation of the star. The period T of rotation is found by measuring the time between pulses. At present, the pulsar in the central region of the Crab nebula has a period of rotation of T=0.033 s, and this period is observed to be increasing at the rate of 1.26 x 10 $^-$$^5$ s/y.

(a) What is the value of the angular acceleration in 2 rad/s$^2$ ?

(b) If its angular acceleration is constant, how many years from now will the pulsar stop rotating?

(c) The pulsar originated in a supernova explosion seen in the year A.D. 1054. What was T for the pulsar when it was born? (Assume constant angular acceleration since then.)

## Problem 4

The oxygen molecule, O$_2$, has a total mass of 5.30 x 10 $^-$$^2$$^6$ kg and a rotational inertia of 1.94 x 10$^-$$^4$$^6$ kg m$^2$ about an axis through the centre of the line joining the atoms and perpendicular to that line. Suppose that such a molecule in a gas has a speed of 500 m s$^-$$^1$ and that its rotational kinetic energy is two-thirds of its translational kinetic energy. Find its angular velocity.

## Problem 5

Two particles, each with mass m, are fastened to each other, and to a rotation axis at O, by two thin rods, each with length l and mass M as shown in the figure. The combination rotates around the rotation axis with angular velocity ω. Obtain algebraic expressions for (a) the rotational inertia of the combination about O and (b) the kinetic energy of rotation about O.

## Problem 6

A uniform thin solid block of mass M and edge lengths a and b is as shown in the figure. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces.

## Problem 7

(a) Show that the rotational inertia of a solid cylinder of mass M radius R about its central axis is equal to the rotational inertia of a thin loop of mass M and radius R/$\sqrt2$ 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 loop about that axis, if the loop has the same mass M and radius k given by

k = $\sqrt{\frac{ I }{M}}$

The radius k of the equivalent loop is called the radius of gyration of the given body. Work in progress, expect frequent changes. Help and feedback is welcome. See discussion page. 