Derive simple harmonic motion from energy equation
Answers
Kinetic energy is the energy possessed by an object when it is in motion. Let’s learn how to calculate the kinetic energy of an object. Consider a particle with mass m performing simple harmonic motion along a path AB. Let O be its mean position. Therefore, OA = OB = a.
The instantaneous velocity of the particle performing S.H.M. at a distance x from the mean position is given by
v= ±ω √a2 – x2
∴ v2 = ω2 ( a2 – x2)
∴ Kinetic energy= 1/2 mv2 = 1/2 m ω2 ( a2 – x2)
As, k/m = ω2
∴ k = m ω2
Kinetic energy= 1/2 k ( a2 – x2) . The equations Ia and Ib can both be used for calculating the kinetic energy of the particle.
Learn how to calculate Velocity and Acceleration in Simple Harmonic Motion.
Potential Energy(P.E.) of Particle Performing S.H.M.
Potential energy is the energy possessed by the particle when it is at rest. Let’s learn how to calculate the potential energy of a particle performing S.H.M. Consider a particle of mass m performing simple harmonic motion at a distance x from its mean position. You know the restoring force acting on the particle is F= -kx where k is the force constant.
Now, the particle is given further infinitesimal displacement dx against the restoring force F. Let the work done to displace the particle be dw. Therefore, The work done dw during the displacement is
dw = – fdx = – (- kx)dx = kxdx
Therefore, the total work done to displace the particle now from 0 to x is
∫dw= ∫kxdx = k ∫x dx
Hence Total work done = 1/2 K x2 = 1/2 m ω2x2
The total work done here is stored in the form of potential energy.
Therefore Potential energy = 1/2 kx2 = 1/2 m ω2x2
Equations IIa and IIb are equations of potential energy of the particle. Thus, potential energy is directly proportional to the square of the displacement, that is P.E. α x2.