Question

A disc is rolling (without slipping) on a horizontal surface. C is its centre and Q and P are two points equidistant from C. Let $v_P$, $v_q$ and $v_c$ be the magnitude of velocities of points P,Q and C respectively, then

• A. $v_Q > v_C > v_P$
• B. $v_Q < v_C < v_P$
• C. $v_Q=v_P,v_C=\frac{1}{2}v_P$
• D. $v_Q < v_C > v_P$

Answer 1

The Correct Option is A

The problem involves a disc rolling without slipping on a horizontal surface. In such a scenario, we need to determine the relationship between the velocities of specific points on the disc: the center of the disc (C), and two equidistant points from the center, P and Q.

To solve this, we start by understanding the movement involved:

1. v_c is the velocity of the center of the disc, which moves horizontally with a speed equal to the translational velocity of the disc.
2. Points on the disc not at the center have velocities dependent on both the translational and rotational motion of the disc. For a disc rolling without slipping, every point has a motion that is a combination of the translational motion of the center of mass and rotation around the center.

For a point P (or Q), which is at a vertical distance equal to the radius (R) away from the center:

1. The point P is moving due to the rotation of the disc but not contributing to the velocity of the center. Thus, its velocity is due just to the rotation component.
2. The point Q is located in the direction of motion (i.e., preceding the point C in the direction of motion), gaining an additional velocity vector due to the rotation of the disc around C.

Analyzing the vectors:

• The magnitude of the velocity at P is smaller than the velocity at the center C because it is largely counteracted by the rotational movement of the disc.
• The magnitude of the velocity at Q is greater than the velocity at C because point Q benefits from the translational velocity due to the motion of the center, plus additional velocity from the rotational motion. Thus, it moves faster than the center.

Considering the above conditions, we conclude the relationship between these velocities: v_Q > v_C > v_P.

Hence, the correct option is: v_Q > v_C > v_P.