Statics of a sphere on two inclined planes A smooth sphere rests in equilibrium, touching two smooth planes BA and BC that are inclined at 45° and 60° respectively to the horizontal. Comparing the support reactions on the two planes, which plane experiences the larger normal reaction?

Introduction / Context:
This problem tests 2D statics of a smooth body in contact with two smooth planes at different inclinations. A sphere supported by two planes experiences only normal reactions at the contacts (no friction), along with its weight acting vertically downward. The task is to compare reaction magnitudes on planes set at 45° and 60° to the horizontal.

Given Data / Assumptions:

• Sphere is smooth; planes are smooth (reactions are normal to the planes).
• Plane BA is inclined at 45° to the horizontal.
• Plane BC is inclined at 60° to the horizontal.
• Weight W acts vertically downward through the sphere’s center.
• Static equilibrium: sum of forces in horizontal and vertical directions equals zero.

Concept / Approach:
For a plane at angle α to the horizontal, the normal reaction makes an angle α to the vertical. Resolve each reaction into vertical and horizontal components and apply equilibrium. Horizontal components of the two reactions must balance each other; vertical components must add up to W. From the horizontal balance, a ratio between the reactions is obtained, which identifies which reaction is larger.

Step-by-Step Solution:

Verification / Alternative check:
As the plane becomes flatter (smaller inclination angle), its normal is closer to the vertical, so a larger reaction is needed to supply sufficient horizontal balance. Since 45° < 60°, the 45° plane supplies the larger reaction, consistent with R_A > R_C.

Why Other Options Are Wrong:

• Less on BA than on BC: Contradicts the ratio R_A/R_C = √3/√2.
• Equal reactions: Only possible for equal inclinations, not here.
• Zero on BA: The sphere would fall; impossible in equilibrium.
• None of these: Incorrect because there is a determinate comparison.

Common Pitfalls:
Confusing the angle the normal makes with horizontal versus vertical, or assuming reactions depend only on vertical components without checking horizontal balance.

Final Answer:
More on BA than on BC