Question

An inclined plane is sliding, and accelerating, on a horizontal frictionless surface. There is a block at rest on the sloping surface, held in place by static friction through the horizontal acceleration of the system. The coefficient of static friction between the block and the inclined plane is 0.615. The slope of the incline plane is 40.5 degrees with respect to the horizontal.
What is minimum acceleration of the inclined plane for the square block not to slide? What is maximum acceleration of the inclined plane for the square block not to slide?

Answer 1

To solve the problem you must make the free body diagram of the block, i'm using a convenient reference system since the majority of forces are acting on that sense. The acceleration has now components on the new reference system. Then we state the x and y axis equations.

We put the friction force as a function of the normal force and friction coefficient.

We solve for the acceleration using both equations and obtain an acceleration of 1.53 m/s² to the left, since the sign is changed meaning that we assumed an opposite direction

That's for the maximum acceleration, the minimum acceleration is zero, since if the system is not in motion or it's not accelerated the block will remain on the plane as a consecuence of the static friction