Question

A uniform rod AB is in equilibrium when resting on a smooth groove, the walls of which are at right angles to each other as shown in the figure. What is the relation between \( \theta \) and \( \phi \) in degrees?

• A. \( \theta = 45^\circ + \phi \)
• B. \( \theta = 45^\circ - \phi \)
• C. \( \theta = 90^\circ - \phi \)
• D. \( \theta = 90^\circ + \phi \)

Hint:

In problems involving rods in contact with perpendicular walls, always look for right-angle triangles and apply angle sum properties.

Answer 1

The Correct Option is C

Step 1: Understand the configuration.
The rod rests in a smooth groove with vertical and horizontal walls at a right angle. The smoothness implies only normal reactions act at contact points. Since the rod is uniform and in equilibrium, its weight \(W\) acts vertically downward from the midpoint \(G\). Step 2: Analyze the geometry.
At equilibrium, the triangle formed by the groove and the rod suggests the angle between the rod and the horizontal wall is \( \phi \), and the angle between the rod and the vertical wall is \( \theta \). From the geometry: \[ \theta + \phi = 90^\circ \quad \Rightarrow \quad \theta = 90^\circ - \phi \]