Question

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle $ \theta $ without slipping and slipping down the incline without rolling is

• A. 5 : 7
• B. 2 : 3
• C. 2 : 5
• D. 7 : 5

Answer 1

The Correct Option is A

To determine the ratio of the accelerations for a solid sphere rolling down an incline without slipping and for the sphere slipping down the incline without rolling, we need to apply the concepts of rotational motion and dynamics.

1. Rolling without slipping:

When the sphere rolls without slipping, both translational and rotational motions are involved. The forces acting on the sphere are gravitational force, normal force, and frictional force.

The net force along the incline determines the translational acceleration \(a\) and can be given by the equation:

F_{\text{net}} = mgsin\theta - f

Where \(f\) is the frictional force, which also provides the torque for rotation. Therefore, by considering the rotational dynamics, we have:

f \cdot R = \frac{2}{5}mRa

Solving for \(f\):

f = \frac{2}{5}ma

Substituting back into the net force equation:

mgsin\theta - \frac{2}{5}ma = ma

\Rightarrow mgsin\theta = \frac{7}{5}ma

\Rightarrow a = \frac{5}{7}gsin\theta

2. Slipping down the incline:

In the case of slipping, there is no rotational motion, hence friction is not required to relate translational and rotational motion. The only force causing acceleration is the component of gravity.

F_{\text{net}} = mgsin\theta = ma\

\Rightarrow a = gsin\theta

3. Ratio of Accelerations:

Now, the ratio of the accelerations is:

\frac{\frac{5}{7}gsin\theta}{gsin\theta} = \frac{5}{7}

Hence, the ratio of the accelerations is 5:7.

Thus, the correct answer is 5 : 7.