Question

A bar AB of weight W rests like a ladder with upper end A against a smooth vertical wall and the lower end B on a rough horizontal plane. If the bar is just on the point of sliding, then the reaction at A is equal to (given that $\mu$ is coefficient of friction) :

• A. $\mu \, W $
• B. $W$
• C. $\frac{W}{\mu}$
• D. None of these

Answer 1

The Correct Option is A

To determine the reaction at point A for a bar AB resting against a smooth vertical wall and a rough horizontal plane, we need to analyze the forces acting on the bar. The bar is on the verge of sliding, which means it is in a state of equilibrium, but any additional force will cause it to move.

Let's break down the forces acting on the bar:

1. The weight of the bar W acts vertically downward at the center of the bar.
2. The normal reaction at the wall, R_A, acts horizontally at point A.
3. The normal reaction from the floor, R_B, acts vertically upward at point B.
4. The frictional force f at point B acts horizontally opposite to R_A.

Since the bar is on the point of sliding, the maximum static friction is acting, which can be expressed as:

f = \mu R_B

where \mu is the coefficient of friction.

For equilibrium, the following conditions must be satisfied:

1. Horizontal forces balance: R_A = f = \mu R_B
2. Vertical forces balance: R_B = W (as the normal force must support the weight of the bar)

From the horizontal force balance equation, we substitute R_B = W:

R_A = \mu W

Thus, the reaction at A is \mu W, meaning the correct answer is \mu W.

Let's rule out other options:

• W: This would imply no contribution from friction, which is incorrect as the bar is on the verge of sliding.
• \frac{W}{\mu}: This would only be relevant if the wall had a frictional force component, which it doesn't.
• None of these: We found that \mu W is indeed correct.