An object (point mass) is projected up an inclined plane from the bottom with an initial speed of 5.0 m/s. The plane is 0.75 m high and the coefficient of kinetic friction between the object and the plane is 0.25.

For what angle of inclination would the mass just come to rest at the top of the plane?

To find the angle of inclination at which the mass just comes to rest at the top of the plane, we need to consider the forces acting on the object.

1. First, let's analyze the forces acting parallel to the inclined plane.
- The gravitational force acting downwards can be decomposed into two components: one parallel to the inclined plane and one perpendicular to it. The component parallel to the inclined plane is m * g * sinθ, where m is the mass of the object and g is the acceleration due to gravity (9.8 m/s^2).
- The kinetic friction force acting in the opposite direction is given by μ * N, where μ is the coefficient of kinetic friction and N is the normal force. The normal force can be calculated as m * g * cosθ.

2. Next, let's analyze the forces acting perpendicular to the inclined plane.
- The component of the gravitational force acting perpendicular to the inclined plane is m * g * cosθ.
- There is no acceleration or force acting in the perpendicular direction.

3. When the object comes to rest, the net force acting on it in the parallel direction is zero, while the net force acting on it in the perpendicular direction is m * g * cosθ.

Now, let's set up the equations to find the angle of inclination (θ) at which the mass just comes to rest:

In the parallel direction:
m * g * sinθ - μ * m * g * cosθ = 0

In the perpendicular direction:
m * g * cosθ = 0

Simplifying the equations:
sinθ - μ * cosθ = 0
cosθ = 0

For the second equation, cosθ = 0, we can see that it is not possible because it would mean the object is not in contact with the surface of the inclined plane. Therefore, we only need to solve the first equation:

sinθ - μ * cosθ = 0

Plugging in the given values:
sinθ - 0.25 * cosθ = 0

Now, to find the angle of inclination (θ), we can rearrange the equation:
θ = arctan(μ)

Plugging in the value of μ:
θ = arctan(0.25)

Using a scientific calculator, we can find the angle of inclination to be approximately 14.04 degrees.

So, the angle of inclination at which the mass would just come to rest at the top of the inclined plane is approximately 14.04 degrees.