A box of mass 40 kg is pulled up by a ropeat a constant velocity on a rough inclinedplane which makes an angle of 30° with[ 4 ] BPHCT-131the horizontal. Determine the coefficient ofkinetic friction between the box and thesurface of the plane. It is given that thetension in the rope is 250 N. Draw the freebody diagram. Take g = 10 2ms−

To determine the coefficient of kinetic friction between the box and the surface of the plane, we can follow these steps:

1. Draw the free body diagram:

   • Draw a box on an inclined plane, making an angle of 30° with the horizontal.
   • Label the forces acting on the box: the weight (mg) acting vertically downwards, the normal force (N) acting perpendicular to the plane, and the frictional force (f) acting parallel to the plane. The tension in the rope (T) is also acting on the box, but it is not relevant for this calculation.

2. Identify the forces:

   • The weight of the box (mg) can be calculated as the mass (40 kg) multiplied by the acceleration due to gravity (10 m/s^2), giving us a weight of 400 N.
   • The normal force (N) can be calculated as the component of the weight perpendicular to the plane, which is N = mg * cos(30°).
   • The frictional force (f) can be calculated as the coefficient of kinetic friction (μ) multiplied by the normal force (N), which is f = μN.

3. Set up the equation:

   • Since the box is moving at a constant velocity, the net force acting on it is zero. Therefore, the sum of the forces in the horizontal direction is equal to zero.
   • The forces in the horizontal direction are the frictional force (f) and the component of the weight parallel to the plane, which is mg * sin(30°).
   • Setting up the equation: f = mg * sin(30°).

4. Solve for the coefficient of kinetic friction:

   • Substitute the values into the equation: μN = mg * sin(30°).
   • Substitute the expressions for N and f: μ * mg * cos(30°) = mg * sin(30°).
   • Simplify the equation: μ * cos(30°) = sin(30°).
   • Divide both sides by cos(30°): μ = sin(30°) / cos(30°).
   • Calculate the value: μ ≈ 0.577.

Therefore, the coefficient of kinetic friction between the box and the surface of the plane is approximately 0.577.