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

Problem 1
How do you determine Rotational Inertia of Geometrical Bodies?

[[media:mec0001.pdf|Solution to problem 1]]

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.

[[media:mec0002.pdf|Solution to problem 2]]

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.)

[[Media:mec003.pdf | Solution to Problem 3]]

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.

[[media:mec0004.pdf|Solution to problem 4]]

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.



[[media:mec0005.pdf|Solution to problem 5]]

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.



[[media:mec0006.pdf|Solution to problem 6]]

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.

[[media:mec0007.pdf|Solution to problem 7]]