A trunk of mass m = 0.7 kg is pushed a distance d = 24 cm up an incline with an angle of inclination theta = 19.0 o by a constant horizontal force P = 400 N (see figure). The coefficient of kinetic friction between the trunk and the incline is 0.12.
Calculate the work done on the trunk by the applied force P.
Calculate the work done on the trunk by the frictional force.
Calculate the work done on the trunk by the gravitational force.
To solve this problem, we need to first decompose the forces acting on the trunk. The forces we have are the applied force P, the frictional force, and the gravitational force.
1. Work done by the applied force P:
The work done by an applied force is given by the equation W = Fd, where W is the work done, F is the force applied, and d is the distance traveled in the direction of the force.
In this case, the applied force P is acting horizontally, so the distance traveled in the direction of the force is the horizontal displacement, which is given by d = cos(theta) * d.
The work done by the applied force P is therefore W = P * cos(theta) * d.
Plugging in the given values, we have W = 400 N * cos(19.0 o) * 0.24 m = 92.426 J.
2. Work done by the frictional force:
The work done by a frictional force is given by the equation W = μk * N * d, where W is the work done, μk is the coefficient of kinetic friction, N is the normal force, and d is the distance traveled in the direction of the force.
The normal force N is given by N = m * g * cos(theta), where m is the mass of the trunk, g is the acceleration due to gravity, and theta is the angle of inclination.
The distance traveled in the direction of the frictional force is the same as the distance traveled in the direction of the applied force, which is given by d = cos(theta) * d.
Plugging in the given values, we have N = 0.7 kg * 9.8 m/s^2 * cos(19.0 o) = 6.936 N.
And d = cos(19.0 o) * 0.24 m = 0.227 m.
The work done by the frictional force is therefore W = 0.12 * 6.936 N * 0.227 m = 0.185 J.
3. Work done by the gravitational force:
The work done by the gravitational force is given by the equation W = m * g * h, where W is the work done, m is the mass of the trunk, g is the acceleration due to gravity, and h is the vertical displacement.
The vertical displacement h is given by h = sin(theta) * d.
Plugging in the given values, we have h = sin(19.0 o) * 0.24 m = 0.080 m.
The work done by the gravitational force is therefore W = 0.7 kg * 9.8 m/s^2 * 0.080 m = 0.548 J.
Therefore, the answers are:
- The work done on the trunk by the applied force P is 92.426 J.
- The work done on the trunk by the frictional force is 0.185 J.
- The work done on the trunk by the gravitational force is 0.548 J.