Pendulums and Springs: Tension - MCAT
It may be counterintuitive, but in a pendulum, the tension in the string during an oscillation is larger than mg. Here is an example to explain why this is so. Imagine for a moment that you are looking at a weight on a string that’s bouncing, and imagine that you are also looking at a pendulum swinging, but that you’re looking at it from the side, so that it’s swinging toward and away from you. Now let’s imagine that the string length, the masses of the weights, and the spring constant, etc., are all chosen so that the period of the oscillation is the same for both, and that the maximum and minimum heights of each weight is the same. From your perspective, they should look the same. Each weight should rise and fall in lock-step.
Now, when the spring-ball is at the bottom, it is easy to think that the tension on the spring must be greater than mg. If the tension were only mg, then the ball would sit motionless at the bottom. The added potential energy comes from the stretching of the spring, and this potential energy will convert to kinetic energy in lifting the ball up again. When it gets to the top, the potential energy comes from its height. This is how an oscillation works: conservation of energy, which converts from one form, to another, then back. In the case of the spring, the energy converts from potential (height) into kinetic (falling down) to potential (of the spring tension) to kinetic (the ball springing back up) and back to the start. This is a bit easier to understand intuitively than the pendulum, but it’s the same principle.
With the pendulum, let’s imagine that the center of rotation is the center of a clock, where the pendulum swings from eight o’clock, down to six, up to four, and then reverses back to six, and up to eight. At eight o’clock and four o’clock, the ball has no kinetic energy–it pauses motionless for an infinitessimally short moment. All of its energy is potential, from the height. At six o’clock, there is no potential energy. All of its energy is kinetic, as it reaches its maximum speed. At the time that the weight is at the bottom or the arc, the direction of the velocity is perpendicular to its arc. The string has the job of redirecting that kinetic energy to make the ball go up to either 8 or 4 o’clock. To do this, the direction of acceleration is perpendicular to the direction of the motion. This is centripetal force. Now, if you imagine a ball hanging motionless from a string, it has no kinetic energy. The total tension on the string in this situation is just -mg, as it is at equilibrium. The swinging pendulum at the bottom of its arc is not at equilibrium at all. Specifically, it has the same force of gravity as the ball at rest, but since the ball in the pendulum is swinging, it also has the force created by the acceleration–that is, the change in direction of the velocity–that comes at the bottom.
I think the key here is to remember that acceleration is not just the change in the speed, but the change in velocity. When you first hear about speed and velocity, it seems like a nit-pick to say that they are not the same, and that velocity is speed + direction. But a point on a wheel spinning at constant speed still accelerates. When you see that F = m*a, remember that acceleration doesn’t care if it’s a change in speed or a change in direction–it creates a force all the same. Since the pendulum ball changes direction of velocity, it must necessarily be accelerating, and since it has mass, there must be a resultant force. If the only tension on the string when the ball was at the bottom was -mg, then the system would be at equilibium, and the ball would be completely motionless.
