Forces, Motion, and Energy

 How are force, acceleration, velocity, and displacement related?

Often, we use F=ma as our starting point when deriving equations, but when discussing simple harmonic motion we can describe the restoring force as F=-kx. 

Setting these two equal to each other gives ma=-kx, or a∝-x. This shows how acceleration is proportional to displacement and will always point towards equilibrium. 

Velocity, however, is not proportional to displacement since an object doesn't need to be displaced in order to have velocity. For example, in the last post about springs, the spring passed through the equilibrium point where displacement was zero with a high velocity due to the restoring force.

Refer to this diagram for a visual representation:

Pendulums

While we have been using springs for examples up until now, pendulums also exhibit SHM. In a pendulum, the force of gravity is the restoring force. This is because the equilibrium position of the pendulum is resting in the center and when the pendulum is actively swinging the only x component present is the tangential component of gravity pointing towards the center. Similarly to spring-mass systems, acceleration is proportional to negative displacement and velocity is not.

Energy in SHM

Typically, there are two main types of energy that come to attention in simple harmonic motion: kinetic and potential. Looking back at the spring-mass system above, we can see how kinetic energy is zero at each extreme (velocity is 0) but hits its maximum at the equilibrium point. Since there are no external forces acting on the system, this must mean that in a perfect world mechanical energy remains constant; spring potential energy must hit its maximum where kinetic energy is at its minimum and vice versa. This constant swap between forms of energy is common in most types of SHM and is seen in the pendulum too; velocity hits its maximum at the equilibrium point, and zero at each maximum height. The only difference is that instead of spring potential energy, pendulums use gravitational potential energy.

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