Question

A semicircular slot of 10-in. radius is cut in a flat plate which rotates about the vertical AD at a constant rate of 14 rad/s. A small, 0.8-lb block E is designed to slide in the slot as the plate rotates. Knowing that the coefficients of friction are µs = 0.35 and µk = 0.25, determine whether the block will slide in the slot if it is released in the position corresponding to (а) θ = 80°, (b) θ = 40°. Also determine the magnitude and the direction of the friction force exerted on the block immediately after it is released.

Answer 1

Consider the flat plate with semicircular slot as shown in figure below:

Here, the verticalis the axis of rotation and is the distance of the point E from the axis of rotation.

Obtain the distance of the point E from the axis of rotation.

Calculate the normal acceleration of the block at point E.

Here, is the angular velocity of the flat plate about its vertical AD.

Substitute for and 14 rad/s for

Initially assume that the block is at rest position in the slot with respect to the plate.

Draw the free body kinematic diagram of the block.

Here, is the friction force, is the normal force, is the weight, and is the mass of the block.

Consider the force equilibrium along the direction.

Consider the force equilibrium along the direction.

Here, is kept before because will be acting in negative y direction.

(a)

When

Calculate the normal force acting on the block.

Substitute 0.8 lb for for for and for .

Calculate the frictional force acting on the block.

Substitute 0.8 lb for for for and for .

Calculate maximum frictional force to decide whether the block will slide or not.

Here, is the static coefficient of friction.

Substitute 0.35 for and 6.3159 lb for

From the above value, we can conclude that .

Actual frictional force is less than the maximum frictional force. So, the block will stay in its position and doesn’t slide in the slot.

Therefore, the block in the slot when and the magnitude of frictional force acting on the block is which acts at an angle ofin second quadrant.

(b)

When

Calculate the normal force acting on the block.

Substitute 0.8 lb for for for and for .

Calculate the frictional force acting on the block.

Substitute 0.8 lb for for for and for .

Calculate the maximum frictional force.

From the above value, we can conclude that .

Therefore, block E in the slot when .

If block E slides in the slot we can write,

When t =0, the block E will be at rest relative to the plate, thus

Therefore, must be directed tangentially to the slot as shown in figure below:

Consider the equilibrium of forces in x direction for the above figure.

Calculate the normal force acting on the block.

Substitute 0.8 lb for for for and for .

When the block is sliding, calculate the frictional force.

If F and are directed as shown in the figure of previous step, then the block E slides downwards in the slot.

Therefore, the magnitude of frictional force exerted on the block when is acts at an angle of in the second quadrant.