Simple Harmonic Motion: Hooke's Law

0:00 Professor Dave here, let's discuss simple harmonic motion. Sometimes when we examine the motion of an object, it will involve a single action, like a rock falling from a cliff down to the ground. This will probably just happen one time after which the rock will remain at rest. But some motion is periodic, meaning repeated, like the motion of the pendulum on a grandfather clock, or the vibration of a spring. We will refer to this kind of motion as simple harmonic motion. Say we have a spring that is attached to some stationary surface, and on the other end of the spring there is a block of a particular mass. If we pull this block so

0:46 as to expand the spring and then we release the block, it will vibrate back and forth between more compressed and less compressed states. If we assume that the surface of motion is completely frictionless, then the elastic potential energy reaches a maximum when the spring is most or least compressed and the block is changing direction, while kinetic energy is at a maximum when the block is right in the middle and moving the fastest. If we were to graph the position of the block against time it would display sinusoidal behavior, just like a trigonometric sine function, where the block continues to occupy the same positions over a particular period of time, which is why we call this periodic

1:31 motion. x equals 0 at the equilibrium position of the block, where it was at before we pulled it, and assuming zero friction the block will oscillate between positive x and negative x indefinitely. In reality, frictional forces will dampen the spring's activity and it will eventually come to a stop, but for many systems an ideal mass-spring system will be a decent approximation for a real one. Of course every spring is different, some are loose like a slinky and some are stiff like the spring that launches a pinball into play. This factor is represented by the spring constant k, which will be unique to a particular spring. The force that a spring can exert is equal to negative k times the displacement of the object it acts upon, and this

2:18 relationship is called Hooke's law. The negative sign indicates that the force of the spring is always opposite the direction of the movement of the object. When the object compresses the spring it will push out, and when the object stretches the spring it will pull in. In both cases, the spring is attempting to move the object back to its equilibrium position, which is why the force applied by the spring can be called a restoring force. The units on a spring constant will be Newtons per meter, so that when you multiply by some distance x, you get the force that must be applied to compress that spring that far. Let's also note that the elastic potential energy of a spring will be equal to one-half kx squared, where k is

3:05 the spring constant and x is the distance it is stretched or compressed. If at equilibrium the distance is zero, so it has zero elastic potential energy, and the maximum elastic potential energy will occur at the maximum distance from this point on either side of the oscillation. This expression looks similar to the expression for kinetic energy, which is convenient since the two will interchange as the mass oscillates. A pendulum also displays periodic motion since it swings back and forth between the same two positions, although this involves gravitational potential energy rather than elastic potential energy, and we will look at this behavior next. Let's check comprehension.

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