A wheel of moment of inertia 0.500 kg-m² and radius 20.0 cm is rotating about its axis at an angular speed of 20.0 rad/s. It picks up a stationary particle of mass 200 g at its edge. Find the new angular speed of the wheel.

A door is hinged at one end and is free to rotate about a vertical axis (Fig. 7.10). Does its weight cause any torque about this axis? Give reason for your answer.

A wheel in uniform motion about an axis passing through its centre and perpendicular to its plane is considered to be in mechanical (translational plus rotational) equilibrium because no net external force or torque is required to sustain its motion. However, the particles that constitute the wheel do experience a centripetal acceleration directed towards the centre. How do you reconcile this fact with the wheel being in equilibrium? How would you set a half-wheel into uniform motion about an axis passing through the centre of mass of the wheel and perpendicular to its plane? Will you require external forces to sustain the motion?

A uniform sphere of mass m and radius R is placed on a rough horizontal surface (Fig. 7.9). The sphere is struck horizontally at a height h from the floor. Match the following:

A. h = R/2 (i) Sphere rolls without slipping with a constant velocity and no loss of energy.

B. h = R (ii) Sphere spins clockwise, loses energy by friction.

C. h = 3R/2 (iii) Sphere spins anti-clockwise, loses energy by friction.

D. h = 7R/5 (iv) Sphere has only a translational motion, loses energy by friction.

Figure 7.5 shows a lamina in x-y plane. Two axes z and z′ pass perpendicular to its plane. A force F acts in the plane of lamina at point P as shown. Which of the following are true? (The point P is closer to z′-axis than the z-axis.)

Figure 7.4 shows two identical particles 1 and 2, each of mass m, moving in opposite directions with same speed v along parallel lines. At a particular instant, r₁ and r₂ are their respective position vectors drawn from point A which is in the plane of the parallel lines. Choose the correct options:

The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?

A simple pendulum of length l is pulled aside to make an angle θ with the vertical. Find the magnitude torque of the weight w of the bob about the point of suspension. When is the torque zero?

A uniform cube of mass m and side a is placed on a frictionless horizontal surface. A vertical force F is applied to the edge as shown in Fig. 7.8. Match the following (most appropriate choice):

A body is in translational equilibrium under the action of coplanar forces. If the torque of these forces is zero about a point, is it necessary that it will also be zero about any other point?

Suppose the platform with the kid in the previous problem is rotating in anticlockwise direction at an angular speed w. The kid starts walking along the rim with a speed u relative to the platform also in the anticlockwise direction. Find the new angular speed of the platform.