A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.11 Combined translational and rotational motion

Figure 84: A cylinder rolling down a rough incline.

as the cylinder falls is converted into rotational kinetic energy, whereas, in the

latter case, all of the released potential energy is converted into translational

kinetic energy. Note that, in both cases, the cylinder’s total kinetic energy at the
bottom of the incline is equal to the released potential energy.

Let us examine the equations of motion of a cylinder, of mass M and radius

b, rolling down a rough slope without slipping. As shown in Fig. 84 , there are

three forces acting on the cylinder. Firstly, we have the cylinder’s weight, M g,

which acts vertically downwards. Secondly, we have the reaction, R, of the slope,
which acts normally outwards from the surface of the slope. Finally, we have the

frictional force, f, which acts up the slope, parallel to its surface.

As we have already discussed, we can most easily describe the translational

motion of an extended body by following the motion of its centre of mass. This

motion is equivalent to that of a point particle, whose mass equals that of the

body, which is subject to the same external forces as those that act on the body.

Thus, applying the three forces, M g, R, and f, to the cylinder’s centre of mass,

and resolving in the direction normal to the surface of the slope, we obtain

R = M g cos θ. (8.94)

Furthermore, Newton’s second law, applied to the motion of the centre of mass

centre of mass

cylinder

f (^) b
R
M g
slope
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