Question

A sphere is given an initial push so that it starts rolling (without slipping) up an inclined plane. During its climb

• A. the direction of the force of friction on the sphere is up the incline.
• B. the direction of the net force on the sphere is up the incline.
• C. the net torque on the sphere is zero.
• D. the work done by the force of friction on the sphere is negative.

Hint:

For a rolling body decelerating on an incline, friction always acts in the direction that helps slow down both the rotation and translation in a synchronized manner. Here, that direction is up the incline.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem describes a sphere rolling without slipping up an incline after being given an initial push.
We need to determine the behavior of the forces, torque, and work done on the sphere.

Step 2: Key Formula or Approach:
For rolling without slipping:
- The contact point is instantaneously at rest, so the friction force is static.
- The relation between linear acceleration $a$ and angular acceleration $\alpha$ is:
\[ a = \alpha R \]
- Torque about the center of mass is:
\[ \tau = I \alpha \]

Step 3: Detailed Explanation:

• As the sphere moves up the incline, gravity has a component $mg\sin\theta$ down the incline. This component slows down the linear motion.
- Thus, the linear acceleration $a$ is directed down the incline.

• For rolling without slipping to be maintained, as the linear speed $v$ decreases, the angular speed $\omega$ must also decrease in proportion ($v = \omega R$).
- Since the sphere is rotating counter-clockwise (viewed from the side where the incline goes up to the left), to decrease $\omega$, there must be a clockwise torque.

• The force of gravity and the normal force act through the center of mass, so they cannot produce any torque about the center.
- Therefore, the only force that can produce a torque about the center of mass to slow down the rotation is the static friction force $f$.

• To create a torque that opposes the rotation, the static friction force must act up the incline.

• Let us verify this: static friction up the incline creates a torque $\tau = fR$ that decelerates the rotation, which is consistent with the linear deceleration caused by gravity.

• For pure rolling, the instantaneous velocity of the point of contact is zero, so the work done by static friction is zero.



Step 4: Final Answer:
During its climb, the direction of the force of friction on the sphere is up the incline.