In the following figure, what is the minimum coefficient of friction needed between the block and fixed incline so that the system does not move. A B C D

To determine the minimum coefficient of friction needed between the block and fixed incline for the system to remain stationary, we can follow these steps:

1. Break Down the Problem

1. Identify the forces acting on the block on the incline.
2. Analyze the forces in the direction parallel and perpendicular to the incline.

2. Relevant Concepts

1. The force of gravity acting on the block can be divided into two components:
   • Perpendicular to the incline: $F_{\perp} = mg \cos(\theta)$
   • Parallel to the incline: $F_{\parallel} = mg \sin(\theta)$
2. The friction force must counteract the parallel component of the gravitational force to keep the system stationary:
   • Maximum static friction: $F_{f} = \mu_s F_{\perp} = \mu_s mg \cos(\theta)$
3. For the system not to move: $F_{f} \geq F_{\parallel}$

3. Analysis and Detail

Equating the friction force and the parallel component of gravity: $\mu_s mg \cos(\theta) \geq mg \sin(\theta)$

Dividing both sides by $mg$ (assuming $m \neq 0$): $\mu_s \cos(\theta) \geq \sin(\theta)$

Rearranging gives: $\mu_s \geq \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)$

4. Verify and Summarize

The minimum coefficient of friction $\mu_s$ required is equal to the tangent of the incline angle $\theta$. This guarantees that the frictional force can counteract the component of gravitational force along the incline.

Final Answer

$\mu_s \geq \tan(\theta)$

Thus, the minimum coefficient of friction needed between the block and the fixed incline to prevent motion is $\tan(\theta)$.