19.9 Rolling along an incline

An incline is an ideal arrangement to realize accelerated rolling motion. Force due to gravity acts through the center of mass of the rolling body. Component of gravity parallel to incline accelerates the body in translation as it goes down and decelerates the body as it goes up the incline.

We have already discussed the case of force, whose line of action pass through center of mass. For rolling of the body, the friction between rolling body and surface appears such that the condition as laid down by equation of accelerated rolling is satisfied. When a body rolls down, it has linear acceleration in downward direction. The friction, therefore, acts upward to counter sliding tendency as shown in the figure. This friction constitutes an anticlockwise torque providing the corresponding angular acceleration as required for maintaining the condition of rolling (if linear velocity is increasing, then angular velocity should also increase according to equation of accelerated rolling).

Friction here plays a dual role :

• It decelerates translational motion.
• It accelerates rotational motion.

As the body rolls down, linear velocity increases with time such that its angular velocity also increases simultaneously in accordance with equation of rolling,

The linear acceleration of the COM of the rolling body is equal to the component of acceleration due to gravity in x direction,

The fact that accelerations are constant has important implications. The motions (translational and rotational) of rolling body along an incline can be described by the constant acceleration kinematics i.e. by the equations of accelerated motion.

Analysis of rolling along an incline

The analysis of rolling involves applying Newton’s second law for both translational and rotational motion and using equation of accelerated rolling. First of all, we select an appropriate pair of rectangular coordinates such that motion is along the positive direction of the x-coordinate. The various forces acting on the rolling disk are :

• Force of gravity, Mg, acting downward.
• Normal force, N, perpendicular to the incline in y-direction.
• Static friction, $f_s$ , acting upward.

The force/ force components are acting in mutually perpendicular directions. As such, we can analyze motion in x-direction independently.

Thus, confining force analysis in x-direction, we apply Newton's law of motion for linear motion as :

Similarly, we apply Newton's law for rotation :

$$\begin{array}{l}\tau =I\alpha \end{array}$$

We note here that force due to gravity and normal force pass through the center of mass. As such, they do not constitute torque on the rolling disk. It is only the friction that applies torque on the disk, which is given as :

At this stage, we can make use of third equation that connects linear and angular accelerations for rolling without sliding :

Combining equations 4 and 5,

$$\begin{array}{l}{f}_{s}R=I\alpha =I\frac{a_C}{R}\\ \Rightarrow f_s=\frac{I{a}_C}{R^2}\end{array}$$

Substituting the expression of friction as above in the equation – 3,

$$\begin{array}{l}\Rightarrow \mathrm{Mg}\sin \theta -\frac{I{a}_C}{R^2}=M{a}_C\\ \Rightarrow a_C(M+\frac{I}{R^2})=\mathrm{Mg}\sin \theta \\ \Rightarrow a_C=\frac{g\sin \theta }{(1+\frac{I}{M{R}^2})}\end{array}$$

We can use this expression to find linear acceleration (hence angular acceleration also) for circular rolling bodies like ring, disk, cylinder and sphere etc. We only need to use appropriate expression of moment of inertia as the case may be.

There is a caveat in the sign of the acceleration of rolling body down the incline. If we choose the orientation of x - coordinate is opposite direction to the one above, then the acceleration of the rolling body is given by the same expression, but with a negative sign preceding it :

$$\begin{array}{l}{a}_C=-\frac{g\sin \theta }{(1+\frac{I}{M{R}^2})}\end{array}$$

The sign of the quantities should not be a source of concern. We should merely stick with the sign convention and choice of coordinate system. Negative sign essentially, though a source of confusion, does not ever change the physical meaning of the quantities in kinematics.

Summary

1: External force due to gravity on a rolling body acts through center of mass.

2: The component of gravity parallel to incline causes rolling body to slide down.

3: Friction acts opposite to the component of gravity parallel to incline.

4: The acceleration of the center of mass of the rolling body on an incline is given by :

$$\begin{array}{l}{a}_C=\frac{g\sin \theta }{(1+\frac{I}{M{R}^2})}\end{array}$$