Changes in potential and kinetic energy as a pendulum swings.
Encyclopædia Britannica, Inc.NARRATOR: A simple pendulum can be made by attaching a bob or weight to one end of a light rod and then suspending the rod from the opposite end, so that it can swing freely and evenly from side to side. An everyday example of a pendulum can be seen in a grandfather clock. The pendulum controls the movement of the parts inside the clock. Once set into motion, the clock's pendulum swings so regularly that the clock keeps accurate time. The swinging of a pendulum is powered by an ongoing process of storage and transformation of energy. When the weighted end of the pendulum is raised to one side by an outside force, the system is given energy. At this point the energy is stored in a form called potential energy. This means that the system has the potential to do work or to become active thanks here to the weight's position high above the lowest point of its swing. Once the weighted end of the pendulum is released, it will become active as gravity pulls it downward. Potential energy is converted to kinetic energy, which is the energy exerted by a moving object. An active pendulum has the most kinetic energy at the lowest point of its swing when the weight is moving fastest.
An ideal pendulum system always contains a stable amount of mechanical energy, that is, the total of kinetic plus potential energy. As the pendulum swings back and forth, the balance between the two types of energy changes constantly. At some points in its swing, the pendulum has more kinetic energy. At other points, it has more potential energy. Of course, no working system is ideal. No pendulum can swing forever because the system loses energy on account of friction. That's why a grandfather clock has to be rewound every few days, to inject a little energy back into the system.
Energy in a Pendulum
In a simple pendulum with no friction, mechanical energy is conserved. Total mechanical energy is a combination of kinetic energy and gravitational potential energy. As the pendulum swings back and forth, there is a constant exchange between kinetic energy and gravitational potential energy.
Potential Energy
The potential energy of the pendulum can be modeled off of the basic equation
PE = mgh
where g is the acceleration due to gravity and h is the height. We often use this equation to model objects in free fall.
However, the pendulum is constrained by the rod or string and is not in free fall. Thus we must express the height in terms of θ, the angle and L, the length of the pendulum. Thus h = L(1 – COS θ)
When θ = 90° the pendulum is at its highest point. The COS 90° = 0, and h = L(1-0) = L, and PE = mgL(1 – COS θ) = mgL
When the pendulum is at its lowest point, θ = 0° COS 0° = 1 and h = L (1-1) = 0, and PE = mgL(1 –1) = 0
At all points in-between the potential energy can be described using PE = mgL(1 – COS θ)
Kinetic Energy
Ignoring friction and other non-conservative forces, we find that in a simple pendulum, mechanical energy is conserved. The kinetic energy would be KE= ½mv2,where m is the mass of the pendulum, and v is the speed of the pendulum.
At its highest point (Point A) the pendulum is momentarily motionless. All of the energy in the pendulum is gravitational potential energy and there is no kinetic energy. At the lowest point (Point D) the pendulum has its greatest speed. All of the energy in the pendulum is kinetic energy and there is no gravitational potential energy. However, the total energy is constant as a function of time. You can observe this in the following BU Physlet on energy in a pendulum.
If there is friction, we have a damped pendulum which exhibits damped harmonic motion. All of the mechanical energy eventually becomes other forms of energy such as heat or sound.
Mass and the Period
Your investigations should have found that mass does not affect the period of a pendulum. One reason to explain this is using conservation of energy.
If we examine the equations for conservation of energy in a pendulum system we find that mass cancels out of the equations.
KEi + PEi= KEf+PEf
[½mv2 + mgL(1-COSq) ]i = [½mv2 + mgL(1-COSq) ]f
There is a direct relationship between the angle θ and the velocity. Because of this, the mass does not affect the behavior of the pendulum and does not alter the period of the pendulum.