Minimum acceleration of Truck for a Block on Vertical Rough Surface without Slipping

A block on vertical rough surface tend to slip in the down ward direction due to its weight and frictional force acts in the opposite direction.We need to give some minimum acceleration to the truck so that the block on vertical surface will not slip.

Let us consider a block on mass m on the rough vertical surface of a truck. Weight of the block acts in the vertical downward direction and it pulls the block in the downward direction. As it tends to move in the downward direction, frictional force starts acting in the upward direction against the relative motion.

To keep the block on the rough vertical surface with out slipping, we shall give some force to the system. For that reason the truck shall start moving with a force and hence with some acceleration. For the block, truck becomes a non inertial frame as it is moving with a acceleration. If the frame of reference is non inertial, newton laws of motion are not valid. To apply newton laws of motion to the body, we shall imagine a pseudo force on it which is equal to the force acting on the reference frame but in the opposite direction.

This force is not actually existing and we have just imagined on the body and it is a virtual force. This kind of pseudo force is required to apply laws of motion to a body in a non inertial frame. As we have applied this much of force on the body, the impact of non inertial frame is nullified.

This pseudo force is applied by the block on the vertical surface of the body and it acts like a action. The vertical wall applies reaction on the block and it is called normal reaction. Thus in this case normal reaction is nothing but the pseudo force on the block and it is perpendicular to the frictional force.

Frictional force is the product of coefficient of friction and normal reaction and normal reaction in this case is applied pseudo force. Weight of the block shall be compensated by frictional force for the system to be in equilibrium.

By balancing this equation, we can get the minimum acceleration that the block shall have. This is nothing but the acceleration that the truck shall have so that the block kept on the vertical wall is not going to slip from its surface.

The mathematical expression for this acceleration is derived as shown here below.