System of Particles and Rotational Motion — Practice

Question 1

Masses 2 kg and 3 kg are at (1,0) and (4,0) m. Find the centre of mass.

Accepted Answer

X_cm = (2 x 1 + 3 x 4)/(2 + 3) = 14/5 = 2.8 m from the origin.

Question 2

A disc has I = (1/2)MR^2 about its central perpendicular axis. Find I about a parallel axis at the rim.

Accepted Answer

Parallel axis theorem: I = I_cm + Md^2 = (1/2)MR^2 + MR^2 = (3/2)MR^2.

Question 3

A torque of 10 N m acts on a body with I = 5 kg m^2 for 4 s from rest. Find the angular velocity attained.

Accepted Answer

alpha = tau/I = 10/5 = 2 rad/s^2. omega = alpha t = 2 x 4 = 8 rad/s.

Question 4

A diver with I = 15 kg m^2 spins at 3 rad/s tucked, then straightens to I = 25 kg m^2. Find the new angular velocity.

Accepted Answer

Conservation of angular momentum: I1 omega1 = I2 omega2. 15 x 3 = 25 x omega2, so omega2 = 1.8 rad/s.

Question 5

A solid sphere rolls down a 10 m high incline. Find its speed at the bottom (g = 10 m/s^2).

Accepted Answer

Mgh = (1/2)MV^2 + (1/2)I omega^2 with I = (2/5)MR^2 and omega = V/R, giving Mgh = (7/10)MV^2. V = sqrt(10gh/7) = sqrt(1000/7) = 11.95 m/s.

Practice — System of Particles and Rotational Motion