A box of mass m slides down an inclined plane that makes an angle of φ with the horizontal. If the coefficient of kinetic friction is μk, find an expression for the acceleration of the box down the slope.

The acceleration due to gravity is g.

a. g ( cosφ + μksinφ)

b. g ( cosφ - μksinφ)

c. g ( sinφ - μkcosφ )

d. g ( sinφ + μkcosφ )

g force down slope = mg sin phi

friction force up slope = mu mg cos phi

a = F/m = g sin phi - mu g cos phi

= g (sin phi - mu cos phi)

To determine the expression for the acceleration of the box down the slope, we need to consider the forces acting on the box.

1. The gravitational force (mg) can be resolved into two components: one parallel to the incline, which contributes to the acceleration, and one perpendicular to the incline.

The component parallel to the incline is mg sinφ.

2. The frictional force (Ff) opposes the motion of the box and acts in the opposite direction to the component of the gravitational force parallel to the incline. The magnitude of the frictional force is given by Ff = μk N, where N is the normal force.

The normal force (N) can be calculated as N = mg cosφ.

3. The net force (Fnet) is the vector sum of the component of the gravitational force parallel to the incline and the frictional force. Since the acceleration is in the same direction as the component of the gravitational force parallel to the incline, we can write:

Fnet = mg sinφ - μkN.

Substituting N = mg cosφ, we have:

Fnet = mg sinφ - μk(mg cosφ).

Using Newton's second law (Fnet = ma), we can set Fnet equal to ma:

ma = mg sinφ - μk(mg cosφ).

Dividing both sides by m, we get:

a = g sinφ - μk g cosφ.

Rearranging terms, we have:

a = g (sinφ - μk cosφ).

Therefore, the correct expression for the acceleration of the box down the slope is:

c. g (sinφ - μk cosφ).

To find the expression for the acceleration of the box down the slope, we need to consider the forces acting on the box.

1. The gravitational force (mg) can be resolved into two components:
- The component perpendicular to the inclined plane is mg * cosφ.
- The component parallel to the inclined plane is mg * sinφ.

2. The frictional force (f) opposing the motion can be calculated using the formula f = μk * N, where N is the normal force exerted on the box by the inclined plane. The normal force can be calculated as N = mg * cosφ.

Now, let's analyze the forces along the inclined plane:
- The force component parallel to the inclined plane is mg * sinφ, acting downwards.
- The frictional force is μk * mg * cosφ, acting upwards.

Using Newton's second law (F = ma), we can write the equation of motion:
ma = mg * sinφ - μk * mg * cosφ

Dividing both sides by m, we get:
a = g * sinφ - μk * g * cosφ

Comparing this equation with the given options, we can see that the correct expression for the acceleration of the box down the slope is:

d. g ( sinφ + μkcosφ )