Problem Statement: Find the minimum value of theta so that block of mass m does not move on rough surface for any value of applied force F. Coefficient of static friction between the block and the surface is u?

Solution:

Understanding the problem: We have a block of mass m placed on a rough surface. A force F is applied on the block at an angle theta. The coefficient of static friction between the block and the surface is u. We need to find the minimum value of theta so that the block does not move for any value of applied force F.

Free body diagram: Let's draw the free body diagram of the block.

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Here, N is the normal force acting on the block, mg is the weight of the block, and fs is the static frictional force acting on the block in the opposite direction to the applied force F.

Conditions for equilibrium: For the block to be in equilibrium, the net force acting on the block should be zero and the net torque acting on the block should also be zero.

Net force equation: The net force equation can be written as:

F*cos(theta) - fs = 0

Here, F*cos(theta) is the horizontal component of the applied force and fs is the static frictional force acting on the block. Since the block is not moving, fs should be equal to its maximum value, which is u*N. Therefore, the above equation can be written as:

F*cos(theta) - u*N = 0

Net torque equation: The net torque equation can be written as:

F*sin(theta)*h - fs*r = 0

Here, F*sin(theta)*h is the torque due to the applied force and fs*r is the torque due to the frictional force. Since the block is not rotating, the net torque acting on the block should be zero. Therefore, the above equation can be written as:

F*sin(theta)*h - u*N*r = 0

Solving for minimum value of theta: We need to find the minimum value of theta for which the block does not move for any value of applied force F. This means that the static frictional force should be equal to its maximum value at all times. Therefore, we can write:

F*cos(theta) - u*N = 0

N = mg

F*sin(theta)*h - u*mg*r = 0

Solving the above equations for theta, we get:

theta = tan^-1(u*h/r)

Therefore, the minimum value of theta for which the block does not move for any value of applied force F is:

theta_min = tan^-1(u*h/r)

Conclusion: In this problem, we analyzed the forces acting on a block of mass m placed on a rough surface and found the minimum value of theta for which the block does not move for any value of applied force F. The key concept used in this problem is the condition for equilibrium, which states that the net force and net torque acting on a body should be zero for it to be in equilibrium.