Consider the net forces acting on the object causing it to accelerate.

To determine the Force of Gravity in the x-direction, we must break the force of gravity into components and examine the side acting in the x-direction.  Using trigonometric functions we get that 

We know that the force of gravity is equal to mg

According to Newton’s 2nd law the force is equal to the mass times the acceleration of the object.

The force of friction is directly related to  μ (the coefficient of friction)  times the normal force.  In this case the normal force is equal to the y component of the force of gravity.

Therefore 

If we substitute this in our original net force equation

Notice that mass is in each piece of the equation so we can cancel it out.

Now we can rearrange and solve for the coefficient of friction

Since the only force pulling both cars down the slope is the force of gravity, both cars will accelerate at the same rate.

It is possible to prove this mathematically by examining the x-component of the gravitational pull.

We know that the force of gravity is equal to mg

According to Newton’s 2nd law the force is equal to the mass times the acceleration of the object.

Both sides have mass so this cancels out of the equation.

Since this is independent of mass, this will be true for both objects.

According to Newton’s 2nd Law the net force on the object is equal to the mass times the acceleration.  The net force is also equal to the sum of the forces involved.

In the second experiment the horizontal applied force is doubled.  However, this does not have any effect on the friction on the object.  Therefore the new equation is

From this equation we can see that the new acceleration will be more than double the original value.

To examine this numerically, let’s assume that the mass of the box is , the friction force is  and the applied force originally was .

Then, let’s double the horizontal force, keeping the friction force the same.

This value is more than double the original value, confirming our answer.

To determine the net force on a system, it is important to consider all the forces between each object in the same direction.  The direction to be considered is along the line of the rope.  The system is considered to be all the blocks together as they will all move with the same acceleration.

According to Newton’s 3rd Law, the Tension in the rope caused by A pulling on B and B pulling on A is equal in magnitude and opposite in direction.  Therefore these two Tension forces cancel out.

From Newton’s 2nd law we know that the net force on a system is equal to the mass of the system times the acceleration of the system

Therefore

We can calculate the force of gravity from the equation Fg = mg

Therefore

We can rearrange this equation and solve for a

The equation for the force of friction is , where μ is the coefficient of static friction.

The normal force is equal to the mass times acceleration due to gravity, but in the opposite direction (negative of the force of gravity).

Since the problem tells us that the force due to friction is , we can plug these values into our original equation to solve for the coefficient of friction.

The coefficient of friction has no units.

Kinetic friction is never greater than static friction. More force is always requires to overcome static friction than is required to overcome kinetic friction. It can require a large force to initiate motion, causing an initial acceleration by overcoming static friction. Once motion has begun, however, less for is required to maintain the motion due to the principles of Newton's first law and inertia.