A 130-kg crate is initially at rest on a horizontal frictionless surface. An applied force of 400 N to the right is exerted on the crate, causing it to move and cover a distance of 26.0 m. At the end of its path, the crate is found to be moving at a speed of 15.0 m/s. (a) What is the work done by the following forces: (i) weight of the crate, (ii) normal force exerted on the crate by the surface, (iii) applied force. (b) What is the net work done on the crate? (c) What is the change in its kinetic energy?
To find the work done by a force, we use the formula:
Work = Force x Distance x cos(theta)
where theta is the angle between the force and the displacement.
(i) The weight of the crate is the force exerted on it due to gravity. It can be calculated as:
Weight = Mass x Gravity
Given that the mass of the crate is 130 kg and acceleration due to gravity is 9.8 m/s^2, we can calculate the weight:
Weight = 130 kg x 9.8 m/s^2 = 1274 N
Since the crate is initially at rest and remains on a horizontal surface, the angle between the weight force and the displacement is 0 degrees. Therefore, cos(0) = 1.
The work done by the weight of the crate is:
Work = Weight x Distance x cos(0) = 1274 N x 26.0 m x 1 = 33,124 J
(ii) The normal force exerted on the crate by the surface is equal in magnitude and opposite in direction to the weight of the crate. Therefore, the normal force is also 1274 N. The angle between the normal force and the displacement is 180 degrees, so cos(180) = -1.
The work done by the normal force is:
Work = Normal Force x Distance x cos(180) = 1274 N x 26.0 m x -1 = -33,124 J
(iii) The applied force is given as 400 N to the right. The angle between the applied force and the displacement is also 180 degrees, so cos(180) = -1.
The work done by the applied force is:
Work = Applied Force x Distance x cos(180) = 400 N x 26.0 m x -1 = -10,400 J
(b) The net work done on the crate is the sum of the individual work done by each force. We can calculate it by adding the work done by the weight, normal force, and applied force:
Net Work = Work by weight + Work by normal force + Work by applied force
= 33,124 J + (-33,124 J) + (-10,400 J)
= -10,400 J
(c) The change in kinetic energy of the crate is equal to the net work done on it, according to the work-energy theorem:
Change in Kinetic Energy = Net Work
Therefore, the change in kinetic energy of the crate is -10,400 J.