How are amplitude and energy related in SHM?

In simple harmonic motion (SHM), the energy is directly proportional to the square of the amplitude.

In more detail, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. It is an important concept in physics, describing the motion of pendulums, springs, and many other systems. To understand the basics of SHM, including the types of forces involved and the motion's characteristics, one can refer to Basics of SHM.

The total energy of the system in SHM is given by the sum of its kinetic and potential energy. The kinetic energy is maximum when the object is at the equilibrium position and the potential energy is maximum when the object is at the extreme positions, i.e., the amplitude. The potential energy (PE) in SHM is given by the formula PE = 1/2 kA^2, where k is the spring constant and A is the amplitude. This shows that the potential energy is directly proportional to the square of the amplitude. For a deeper understanding of energy in SHM and its conservation, you might find Energy in SHM insightful.

Similarly, the kinetic energy (KE) in SHM can be given by the formula KE = 1/2 m v^2, where m is the mass of the object and v is the velocity. At the equilibrium position, the velocity is maximum and is given by v = Aω, where ω is the angular frequency. Substituting this in the kinetic energy formula, we get KE = 1/2 m (Aω)^2, which simplifies to KE = 1/2 m A^2 ω^2. This shows that the kinetic energy is also directly proportional to the square of the amplitude. For those interested in the various types of waves and their parameters in SHM, the page on Types of Waves and Wave Parameters provide additional context.

Therefore, in simple harmonic motion, both the potential and kinetic energy are directly proportional to the square of the amplitude. This means that if the amplitude of the motion is doubled, the energy in the system will increase by a factor of four. This relationship between amplitude and energy is a fundamental aspect of SHM and is crucial in understanding the behaviour of oscillatory systems. To explore how energy is conserved within these systems, further reading can be done on Energy Conservation in SHM.