Question

A small block is made to slide, starting from rest, along two equally rough circular surfaces from A to B through path 1 and 2. The two paths have equal radii of curvature. The speed of the block at the end of the slide was found to be $v_1$ and $v_2$ for path 1 and 2 respectively. Compare $v_1$ and $v_2$

• A. $v_1 > v_2$
• B. $v_1 < v_2$
• C. $v_1 = v_2$

Answer 1

Answer: (A)

$v_1 > v_2$

Answer 2

The work done by gravity is the same for both paths. The work done by friction is negative and its magnitude is proportional to the integral of the normal force along the path. For a convex path (path 1), the normal force is generally smaller than the component of gravity perpendicular to the tangent, while for a concave path (path 2), the normal force is generally larger than the component of gravity perpendicular to the tangent. This is because the normal force contributes to the centripetal force. For a concave path, the normal force is directed towards the center of curvature (upwards), and for a convex path, it is directed away from the center of curvature (upwards). For the same speed and radius of curvature, the required normal force is larger for the concave path. Since the normal force is generally larger for path 2, the work done by friction is larger for path 2. Therefore, the final kinetic energy for path 2 is smaller than for path 1, resulting in a smaller final speed for path 2.