Guide 8-4. Example of How Changing the System Affects the Solution to an Energy Problem

This guide will examine how changing the system affects the solution to an energy problem. We'll see that the general relationship W_ext = ΔE_sys works no matter how the system is selected.

Problem: A block is sliding up an inclined plane without friction. Given the initial velocity of the block and the angle of inclination of the plane, determine how far the block travels along the plane.

Method 1:  System = Block + Earth

Open this animation to see the situation. The system has been selected to include the block and the Earth. The plane isn't included, because the normal force does no work on the block. Run the animation to see how the energy bars change. Screen captures of the bars are shown below at three instants of time. In Figure 1, all the energy is kinetic, since the ball starts at y_i = 0 for which U_g = 0. In Figure 2, U_g has increased while K has decreased, but the sum, E_sys, remains constant. In Figure 3, the ball is at the top of its path where the velocity and hence kinetic energy are zero. Hence, all the energy is gravitational potential.

Figure 1	Figure 2	Figure 3
t = 0	partway up	highest position

Now here's a complete solution with the system as selected above.

Given: yi = 0, vf = 0, vi, θ Goal:  Find d System:  block + Earth Ext. forces:  normal (does no work) States:  See diagram ΔUg is positive ΔK is negative	yi = 0 is defined to be the 0 level for gravitational potential energy.
	The energy of the system includes kinetic energy and gravitational potential energy. 0 work is done by external forces.   yf = dsinθ

Method 2:  System = Block

This time we select only the block as the system. In order to see this in the animation, enter -1 for the System input, reset, and run. There is no gravitational potential energy bar this time. Without the Earth in the system, the only form of energy included is kinetic. Gravity is a force that is external to the system and therefore does work on the system. This work is negative, because the angle between the gravitational force and the displacement is greater than 90° as shown to the right.

Screen captures of the energy bars are shown below. In every case, the energy of the system equals the kinetic energy of the block, because the block is the only object in the system. Thus, as the kinetic energy of the block changes, so does the energy of the system. The latter isn't conserved, because gravity, which is a force external to the system, does work on the block. This work is negative as shown in Figures 5 and 6. Note that the change in kinetic energy from the initial state is always equal to the work done by gravity.

Figure 4	Figure 5	Figure 6
t = 0	partway up	highest position

Now here's a complete solution with the system as selected above.

Given: yi = 0, vf = 0, vi, θ Goal:  Find d System:  block Ext. forces:  normal (does no work) gravity (does work) States:  See diagram ΔK is negative
	The energy of the system includes kinetic energy only. Work is done by gravity.

 

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