A block of mass m is pressed against a vertical wall with a force F acting at an angle theta. What are the maximum and minimum forces of F that can be applied and still have the block remain stationary?
If theta is the angle above horizontal and there is no friction
then
F sin theta must be = m g
if theta = 0 , F is infinite
if theta = 90 degree, F = m g
oops sorry I forgot to mention express it in terms of m, g, theta, and us ( coefficient of static friction)
to prevent sliding down:
F sin T >/= m g - mu m g cos T
to prevent sliding up
F sin T </= m g + mu m g cos T
To determine the maximum and minimum forces of F that can be applied for the block to remain stationary, we need to consider the forces acting on the block.
First, let's analyze the forces acting on the block:
1. Normal Force (N): The force exerted by the wall on the block perpendicular to the wall's surface.
2. Force of Gravity (mg): The force pulling the block downward due to its mass.
3. Applied Force (F): The force applied on the block at an angle theta against the wall.
To keep the block stationary, two conditions need to be satisfied:
1. The applied force F must not exceed the maximum static friction force (Ff_max) between the block and the wall. If F exceeds Ff_max, the block will start sliding.
2. The applied force F must not be less than the minimum static friction force (Ff_min) in order to prevent the block from sliding downward.
To find the maximum force F_max, we need to calculate Ff_max:
1. Determine the vertical component of the applied force:
F_vertical = F * sin(theta)
2. Calculate the maximum static friction force:
Ff_max = µ * N
where µ is the coefficient of static friction between the block and the wall, and N is the normal force exerted by the wall on the block. N can be calculated as N = mg*cos(theta), where m is the mass of the block and g is the acceleration due to gravity.
Therefore, F_max = Ff_max = µ * mg * cos(theta)
Similarly, to find the minimum force F_min, we need to calculate Ff_min:
1. Determine the horizontal component of the applied force:
F_horizontal = F * cos(theta)
2. Calculate the minimum static friction force:
Ff_min = µ * N
where N is the same as above, N = mg*cos(theta).
Therefore, F_min = Ff_min = µ * mg * cos(theta)
Remember that the coefficient of static friction depends on the surfaces in contact. If the surfaces are not specified, you will need additional information to determine the coefficient of static friction (µ) and thus the maximum and minimum forces.