A 4.0 kg box is raised from rest a distance of 6.0 m by a vertical force of 110 N.
(a) Find the work done by the force.
(b) Find the work done by gravity.
(c) Find the final kinetic energy of the box.
To find the answers to these questions, we need to use the concepts of work, gravitational potential energy, and kinetic energy.
(a) Work done by a force is given by the formula:
Work = Force * Distance * cos(theta)
where theta is the angle between the force and displacement vectors. In this case, the force is acting vertically upwards and the displacement is also vertical. So, theta = 0, and cos(theta) = 1. Thus, the work done by the force can be calculated as:
Work = 110 N * 6.0 m * cos(0)
Work = 660 Joules
(b) The work done by gravity can be calculated using the formula:
Work = Force of gravity * vertical displacement
The force of gravity can be calculated using the formula:
Force of gravity = Mass * Acceleration due to gravity
Force of gravity = 4.0 kg * 9.8 m/s^2
Force of gravity = 39.2 N
So, the work done by gravity is:
Work = 39.2 N * 6.0 m
Work = 235.2 Joules
(c) The final kinetic energy of the box can be calculated using the formula:
Kinetic Energy = 1/2 * Mass * Velocity^2
The final velocity of the box can be calculated using the concept of conservation of energy, which states that the total work done on an object is equal to the change in its energy. In this case, the total work done on the box is equal to the sum of the work done by the force and the work done by gravity.
Total work = Work by force + Work by gravity
Total work = 660 Joules + 235.2 Joules
Total work = 895.2 Joules
The change in energy of the box is given by the work done by force, which is equal to the final kinetic energy.
895.2 Joules = 1/2 * 4.0 kg * Velocity^2
Simplifying the equation, we get:
Velocity^2 = 895.2 Joules * 2 / 4.0 kg
Velocity^2 = 447.6 m^2/s^2
Taking the square root of both sides, we get:
Velocity = sqrt(447.6 m^2/s^2)
Velocity ≈ 21.15 m/s
Therefore, the final kinetic energy of the box is approximately:
Kinetic Energy = 1/2 * 4.0 kg * (21.15 m/s)^2
Kinetic Energy ≈ 887.3 Joules