Question

Learning Goal:

To understand how to apply the law of conservation of energy to situations with and without nonconservative forces acting.

The law of conservation of energy states the following:

In an isolated system the total energy remains constant.

If the objects within the system interact through gravitational and elastic forces only, then the total mechanical energy is conserved.

The mechanical energy of a system is defined as the sum of kinetic energy K and potential energy U. For such systems in which no forces other than the gravitational and elastic forces do work, the law of conservation of energy can be written as

Ki+Ui=Kf+Uf,

where the quantities with subscript "i" refer to the "initial" moment and those with subscript "f" refer to the final moment. A wise choice of initial and final moments, which is not always obvious, may significantly simplify the solution.

The kinetic energy of an object that has mass m and velocity v is given by

K=12mv2.

Potential energy, in contrast, has many forms. Two forms that you will be dealing with often are gravitational and elastic potential energy. Gravitational potential energy is the energy possessed by elevated objects. For small heights, it can be found as

Ug=mgh,

where m is the mass of the object, g is the acceleration due to gravity, and h is the elevation of the object above the zero level. The zero level is the elevation at which the gravitational potential energy is assumed to be zero. The choice of the zero level is dictated by convenience; typically (but not necessarily), it is selected to coincide with the lowest position of the object during the motion explored in the problem.

Elastic potential energy is associated with stretched or compressed elastic objects such as springs. For a spring with a force constant k, stretched or compressed a distance x, the associated elastic potential energy is

Ue=12kx2.

When all three types of energy change, the law of conservation of energy for an object of mass m can be written as

12mv2i+mghi+12kx2i=12mv2f+mghf+12kx2f.

The gravitational force and the elastic force are two examples of conservative forces. What if nonconservative forces, such as friction, also act within the system? In that case, the total mechanical energy will change. The law of conservation of energy is then written as

12mv2i+mghi+12kx2i+Wnc=12mv2f+mghf+12kx2f,

where Wnc represents the work done by the nonconservative forces acting on the object between the initial and the final moments. The work Wnc is usually negative; that is, the nonconservative forces tend to decrease, ordissipate, the mechanical energy of the system.

In this problem, we will consider the following situation as depicted in the diagram: (Figure 1) A block of mass m slides at a speed v along a horizontal smooth table. It next slides down a smooth ramp, descending a height h, and then slides along a horizontal rough floor, stopping eventually. Assume that the block slides slowly enough so that it does not lose contact with the supporting surfaces (table, ramp, or floor).

You will analyze the motion of the block at different moments using the law of conservation of energy.

Using conservation of energy, find the speed vb of the block at the bottom of the ramp.

Express your answer in terms of some or all the variables m, v, and h and any appropriate constants.

Answer 1