A box is pushed a distance d along a ramp at a constant speed by a horizontal force, Fa. The mass of the
box is m. The ramp is inclined at an angle of � above the horizontal and the coe�cient of kinetic friction
between the box and the ramp is �k.
To determine the acceleration of the box sliding down the ramp under the influence of the horizontal force Fa, we need to consider the forces acting on the box:
1. The force of gravity, which can be broken down into two components:
- The component perpendicular to the ramp, mg * sin(θ), where θ is the angle of inclination.
- The component parallel to the ramp, mg * cos(θ).
2. The normal force, which is the force exerted by the ramp on the box perpendicular to the plane. This force counteracts the perpendicular component of gravity and is equal to mg * cos(θ).
3. The force of kinetic friction, which opposes the direction of motion and is given by the equation fk = μk * N, where μk is the coefficient of kinetic friction and N is the normal force.
4. The horizontal force Fa, which is pushing the box along the ramp.
Since we are given that the box is moving at a constant speed, this implies that the acceleration is zero. Therefore, the sum of the forces acting on the box in the horizontal direction is zero.
The horizontal forces on the box can be represented as follows:
Fa - fk = 0
Substituting the equation for fk, we get:
Fa - μk * N = 0
Multiplying N by cos(θ) to account for the vertical component of N, we have:
Fa - μk * mg * cos(θ) = 0
Rearranging the equation, we can solve for Fa:
Fa = μk * mg * cos(θ)
Therefore, the horizontal force required to push the box along the ramp at a constant speed is Fa = μk * mg * cos(θ).