Students should be familiar with the general properties of simple harmonic motion (SHM). Students should also be familiar with the concepts of kinetic and potential energy and energy conservation.
Students will learn how to use energy conservation to obtain information about the potential energy of a weighted spring. They will be able to describe the time dependence of the potential energy when the spring is oscillating. They will learn that the energy of the weighted spring is proportional to the square of the amplitude of oscillation of the spring.
Students should understand the applet functions that are described in Help and ShowMe. The applet should be open. The step-by-step instructions on this page are to be done in the applet. You may need to toggle back and forth between instructions and applet if your screen space is limited.
| The kinetic energy KE
of weighted spring varies in time when the weight is oscillating up and
down.
Does the kinetic energy of a weighted spring depend on time t? If so, how? Use the applet to find out. On the applet,
Figure 1
|
Sketch the position (y) vs. time (t) and velocity (vy) vs. time (t) graphs below. On the graphs, mark the points where the spring passes its top (T), bottom (B), and equilibrium positions (E).
Position vs. Time
Velocity vs. Time
Based on the velocity vs. time graph, is the kinetic energy of the weighted spring changing with time? Explain.
Sketch the kinetic energy KE of the oscillating spring vs. time by selecting
it from the graph menu ( Kinetic Energy vs. Time Does your kinetic energy graph support your answer from Exercise 2? |
| The Kinetic Energy of a spring is related to its mass (m) amplitude (A), angular frequency (w) and time (t). | ||||||||||||||||||
| Expressed as an equation,
|
Rewind the applet, and step through the motion until the weight is as close to its equilibrium point as possible (the energy column would be entirely red at the equilibrium point). Record the value of the kinetic energy at this point from the Data box.
KE = _____________ J
Demonstrate how this value is calculated using equation (1).
The potential energy of the simple pendulum is strictly speaking the gravitational potential energy of the pendulum-earth system. We will refer to it as the "potential energy of the pendulum" for short.
The mechanical energy E, energy for short, of the simple pendulum stays constant during the pendulum's motion (if the motion is undamped, as it is in the applet). One says the energy is conserved. The mechanical energy is the sum of kinetic and potential energy,
E = KE + PE (2)
On the applet, play the motion and observe how there is an ongoing conversion of potential into kinetic energy or vice versa during the motion even though the sum of the two energies remains constant. This is observed by selecting the potential energy, kinetic energy, and mechanical energy from the graph menu. Sketch all three energy lines on single graph. Energy vs. Time
|
Now we will use equation (2) together with equation (1) to obtain information about the time dependence of the potential energy.
Reset (
Sketch a graph of the potential energy as function of time. Mark the points where the weight passes the top (T), bottom (B), and equilibrium positions (E). Potential Energy vs. Time
|
The potential energy at the bob's far right and far left positions and the value of the mechanical energy are calculated as follows.
The potential energy of the spring is 0 when the weight is at its equilibrium position. Therefore, at this point, the mechanical energy E is equal to the kinetic energy KE (all the energy at the equilibrium position is in the form of kinetic).
When the weight is either at the top or bottom positions, it is momentarily at rest and therefore its kinetic energy equals zero. At these two positions, all the energy is in the form of potential.
Since E is constant during the motion, the potential energy at the top or bottom is equal to the kinetic energy when the weight is passing its equilibrium position,
PEtop,bottom = E = KEmax (3)
Equation (3) combined with equation (1) for the kinetic energy, is used to derive a general expression for the potential energy at the top or bottom of the motion and for the energy E.
| Potential Energy The potential energy of a weighted spring is related to its mass (m) amplitude (A), angular frequency (w). | |||||||||||||||
| Expressed as an equation,
|
Use equation (4) to calculate the values of PEtop,bottom and E for the applet settings in Exercise 6. Verify your answer with the values in the Data box and also with the value of the kinetic energy near the equilibrium point determined in Exercise 4 of the previous section. Note: the latter may not be quite equal to KEmax if you did not stop the motion at the exact equilibrium point.
There is one other useful equation for determining the potential energy at the maximum amplitude. The angular frequency w is related to the spring constant (k) and the mass of the weight (m).
(5)
Substituting this expression for ω2 into equation (4) gives an expression for the energy in terms of k and A.
(6)
Substitute the applet settings into equation (6) and check if the resulting value of PEtop,bottom agrees with that obtained in Exercise 7.
Physics 20-30 v1.0
©2004 Alberta Learning (www.learnalberta.ca)
Last Updated: June 16, 2004
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