Question 19.CE.7: Energy of a swing at different locations A child sits on a s......
Energy of a swing at different locations
A child sits on a swing that hangs straight down and is at rest. Draw energy bar charts for the child+swing+Earth system; (a) as a person pulls the child back in preparation for the first swing; (b) at the moment the person releases the swing while it is at its elevated position; (c) as the swing passes the equilibrium position; (d) when it reaches half the maximum height on the other side; and (e) as it passes the equilibrium position moving in the opposite direction.
Sketch and translate Start with the sketches of the process at the required instants (see the left sides of the accompanying figures).
Simplify and diagram We ignore air resistance and friction at the ropes’ pivot points. Choose the zero level of gravitational potential energy at the position of the seat when the swing is vertical. The ropes are always perpendicular to the child’s motion and consequently do zero work on the system. The only external object that does work on the system is the person who raises the swing to its starting position. We can now represent the process with bar charts (see the right sides of the figures). Note that the first chart shows work done on the system. Then energy switches between all gravitational potential energy at x=\pm A and all kinetic energy when passing through the equilibrium position at x = 0 to half gravitational and half kinetic at position (d).
Try it yourself: Suppose that you do not ignore air resistance and friction and that after several swings the child returns to only half the original maximum height on the right side. Construct a bar chart for that moment.
Answer: We include the air and the pivots in the system. Half of the original gravitational potential energy has been converted to internal energy due to air resistance and friction at the ropes’ pivots, as shown below.
