Question

A particle of mass m executes SHM represented by the relation: X- A cos (wt). Derive formula for its:

(1) Kinetic Energy (KE)
(2) Potential Energy (PE)
(3) Total Energy when the particle is at position x, in terms of m, a, and x.

Answer 1

The formula for kinetic energy (KE) of a particle in simple harmonic motion (SHM) is 1/2 * m * v^2. The formula for potential energy (PE) of a particle in SHM is 1/2 * k * x^2. The total energy (TE) of a particle in SHM is the sum of its kinetic energy and potential energy.

Step-by-step explanation:

Kinetic Energy (KE):

The formula for kinetic energy (KE) of a particle in simple harmonic motion (SHM) is given by KE = 1/2 * m * v^2, where m is the mass of the particle and v is the velocity of the particle. In SHM, the velocity can be expressed as v = -A * w * sin(wt), where A represents the amplitude of the motion and w is the angular frequency. Substitute this velocity equation into the formula for KE to get a formula for KE in terms of m, A, and w.

Potential Energy (PE):

The formula for potential energy (PE) of a particle in SHM is given by PE = 1/2 * k * x^2, where k is the spring constant and x is the displacement of the particle from its equilibrium position. In SHM, the displacement can be expressed as x = A * cos(wt), where A represents the amplitude of the motion and w is the angular frequency. Substitute this displacement equation into the formula for PE to get a formula for PE in terms of k, A, and w.

Total Energy (TE):

The total energy (TE) of a particle in SHM is the sum of its kinetic energy and potential energy. TE = KE + PE. Substitute the formulas for KE and PE derived earlier, and simplify to get a formula for TE in terms of m, A, and x.