Question

Block A weighs 445N and block B weighs 130N. The coefficient of static friction between the inclined plane and block A is 0.565, and the coefficient of kinetic friction is 0.258. The plane is inclined at the angle θ = 42.0. (a) Find the acceleration of the system if A is initially at rest. (b) Find the acceleration of the system if A is initially moving up the incline. (c) What is the acceleration of the system if A is initially moving down the incline?

Answer 1

The FBD for both bodies are shown below. Friction force depends on the direction of tendency of movement. Let's describe the three cases of the problem.

Assuming acceleration as positive in the x direction, for body A

And for body B

Reeplacing into equation of body A

a.

Since it is on rest, friction force is equal to maximum static friction, then friction coefficient is 0.565. Let's suppose that body A is falling, then acceleration should be negative and the sign of the friction force is positive. With the given data:

This value contraditcs the assumption of body A falling, then this is imposible. Now lets assume it is going up, then friction is negative and acceleration should be positive.

Again, the sign contraditcs the assumption of the movement. In this case, this means that the system is under equilibrium regardless the tendency of movement of body A under static conditions. Another way to demostrate this is by calculating minimum friction to overcome static state.

Since the value of the minimum coefficient is positive, then minimum friction force is positive, which conduts to that the body A will fall in case of overcoming static friction. But in this case, the actual friction coefficient (0.565) is higher thatn minimum (0.5073), then, static state is not overcomed. Finally, acceleration of the system when the body A is under static conditions is 0.

b.

When body A moves up, then friction is kinetic and it has a negative sign:

In this case, the acceleration opposes to the movement (opposite signs). This means that is decelerating, and eventually will stop and stay again in equilibrium, as shown in part a.

c.

When body A moves down, then friction is kinetic and it has a positive sign:

This time, acceleration and movement go in the same direction, which means that body A will accelerate and fall continuously.