The pendulum has a mass of 1 kg and in figure (a) it is moved to a maximum vertical position of 1.8 m. It is released from that position (starts from rest at the top of its motion, highest position) and is allowed to oscillate back and forth. We assume that there is no air resistance or friction in this example, so the pendulum would continue to oscillate forever unless a force was applied to stop it. We also assume that the acceleration due to gravity is 10 m/s2. Fill in the table below for the potential and kinetic energy of the pendulum mass at the various positions of its motion.
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
Let's see if you can apply what you learned about conservation of energy information about potential energy, kinetic energy, and total energy of the system. In the figure below, a simple pendulum is represented at various positions of its motion. The pendulum has a mass of 1 kg and in figure (a) it is moved to a maximum vertical position of 1.8 m. It is released from that position (starts from rest at the top of its motion, highest position) and is allowed to oscillate back and forth. We assume that there is no air resistance or friction in this example, so the pendulum would continue to oscillate forever unless a force was applied to stop it. We also assume that the acceleration due to gravity is 10 m/s2. Fill in the table below for the potential and kinetic energy of the pendulum mass at the various positions of its motion.
Start by calculating the potential energy at the highest position using the fact that PE = mgh. At the highest position, the object is released and thus has no velocity or KE. Remember that due to conservation of energy, the total combined energy of potential + kinetic is equal to the total mechanical energy of the system. Once you know the total energy of the system, E, for the highest position, you also know the total energy for all positions since energy is conserved. For position (b), calculate the potential energy using, PE = mgh. Once you know the PE solve for the kinetic energy according to KE + PE = E. Once you know the kinetic energy of the pendulum, use the fact that KE = 1/2 m v2 to calculate the speed of the pendulum at the position. Repeat this for the remaining positions.
| Figure | PE (J) | + | KE (J) | = | E = PE + KE | speed (m/s) |
| (a) highest position on left = 1.8 m | + | 0 | = | 0 | ||
| (b) half way to lowest position = 0.9 m | + | = | ||||
| (c) lowest position = 0 m | + | = | ||||
| (d)highest position on right side = 1.8 m | + | = |
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