Question

Problem 59a

Answer 1

Given information

To determine: The crate's acceleration as it slides down an inclined plane under the influence of kinetic friction.

Given:

The length of the incline  

The angle at which the incline is inclined  

The coefficient of kinetic friction    

The initial velocity of the crate

Step-by-step explanation

Step 1.

Formula used:

The mass crate slides along down a plane that is inclined to the horizontal. The crate's free body diagram is illustrated below.

The weight  of the crate acts vertically downwards. The normal force acts perpendicular to the inclined plane. The weight is resolved into two components, and  parallel and perpendicular to the inclined plane. The force of kinetic friction  

acts opposite to the direction of motion of the box. There is a net force acting on the box in the downward direction parallel to the inclined plane accelerating it at a rate . Since there is no motion perpendicular to the incline,

 

The normal force is related to the kinetic friction force as follows:

 

Because the box is sliding down the incline, there is a downward net force F. This is supplied by,

 

Now, substitute equations (1) and (2) in (3),

 

         

Now, from Newton’s second law of motion,we know that

 

The resultant acceleration of the crate down the gradient is shown by  ,

Therefore,