Work is determined using the equation . Here,  is the force applied, is the displacement of the object, and  is the angle of the force relative to the movement of the object. Since gravity is acting on the box, we can solve for the force of gravity causing the movement of the box. Note that in this case refers to the angle between gravity and the box's path; thus, the angle will be 60^o, rather than 30^o.

Notice how the work done by gravity is equal to the potential energy of the box at the top of the ramp. This is because mechanical energy is conserved in the system; thus, we can set the two equations equal to each other.

Work is represented by the product of force and displacement:

We are given the force on the box and the distance it travels. Use these values to calculate the work done.

Note that the mass of the box for this problem is irrelevant. There is no vertical displacement, so the force of gravity does not come into play. The only force that matters in this question is the "pushing force."

Work is generally considered a path function. This means that the amount of work depends on the path taken.

Work in a gravitational field, however, is not a path function; it is a state function. This means that work is independent of the path taken in a gravitational field. This occurs because the force due to gravity acts only in one direction (downwards); therefore, the work performed with or against gravity will only depend on the vertical distance travelled.

In this question both students lift the boulder along the same vertical distance (a distance of ). Student Z might have travelled a larger total distance by using the inclined plane, but the smaller  compensates for the larger distance in the work formula.

The work done is equal to the gravitational potential energy of the block after it has been lifted.

The gravitational potential energy is calculated using the following formula:

We are given the mass and the change in height, and we know the acceleration due to gravity. Using these values, we can solve for the change in potential energy by multiplication.

The work done by any force that is perpendicular to the displacement is equal to zero. Since the block is moving horizontally, the net force in the vertical direction will be equal to zero; therefore, work done by either gravity or the normal force will be equal to zero.

Work is given by , the dot product of force and displacement. Since the dot product only sees the components of vectors which are parallel to each other, the dot product of two perpendicular vectors is 0. Force is directed radially inward, while displacement is directed tangent to the circumference. At any point along this object's path, the force is perpendicular to displacement, so  is simply 0.

Remember that the work done by a force on an object is dependent on the angle of the force to the object's displacement. The normal force acts perpendicularly to the ramp, which means it has an angle of 90^o with respect to the box's displacement.

Because , the total work done on the box by the normal force is .

Since the height of the books is constant, there is no change in potential energy. Similarly, since the speed of the books is constant, there is no change in kinetic energy. By the work-energy theorem, work is equal to the change in mechanical energy (including potential and kinetic). If there is zero change in energy, the work done must also be zero.

We can also look at this problem in terms of forces. There is zero acceleration on the books, since they do not move in the vertical direction and the horizontal velocity is constant. If there is no acceleration, then there is no force. If there is no force, then there is no work, according to Newton's second law.

The amount of work performed to get child 1 down the ramp is equal to the amount of kinetic energy gained by her from top to bottom, via the work-energy theorem. Unfortunately, we are not given her kinetic energy. We can't use the 5m/s in the passage, because she stops herself before sliding to the lake surface.

We must instead recognize the potential energy lost is equal to kinetic energy gained. The potential energy that she has lost is given by mgh.

This is an application of the work-energy theorem. The amount of work done by the force of gravity to move child 1 down the ramp is equal to force * distance, as well as equal to the amount of kinetic energy picked up down the ramp.

The force down the ramp is equal to mg*sin(30^o) = 600J * 0.5 = 300J