a) Proof that for any rigid object with mass M that moves in a uniform gravitational eld, the change in gravitational potential energy of the object is the same as that of a point mass M that moves the same as the centre of mass of the object. b} Proof that the torque by a uniform gravitational force on an object relative to a specied point is the same as the gravitational torque on a point mass at the centre of mass of the object, rotating relative to the same point. c] Consider a sphere that rolls without slipping on a ramp whose surface makes an angle ot with the horizontal [Figure 1). Choose the origin 0 at the contact point between the sphere and the ramp surface. Proof that the gravitational torque on the sphere about the contact point 0 induces the same acceleration of the centre ofthe sphere, C, as the component ofgravity parallel to the surface of the ramp. Figure 1. We consider a sphere that rolls on a ramp without slipping as an object that momentarily rotates about an axis through the contact point 0, perpendicular to the drawing. Relative to 0, there is a torque on the object due to the force of gravity (see problem 1b}. In problem lo, we apply the condition that the sphere rolls without slipping to relate the angular acceleration due to this torque to the acceleration of the centre Cofthe sphere. Comment: The motion of an object that rolls without slipping can indeed be described at each time as the object momentarily spinning about the contact point. The kinetic energy of a sphere that rolls without slipping can be written as 1 1 Substitute the no-slip condition em 2 MR so we may write 1 1 1 1 T = Eli/rein\" + 51mm? = 5 (MR2 + 1cm)w2 = in} The last step isjustifled by the parallel axis theorem for rotation about an axis through 0