Question

find out the displacement produced by the particle when the total energy is 1/2 of kinetic and pottential

Answer 1

Potential Energy

 

Let at any instant t, the displacement of particle of mass m from its mean position be x at that constant, the acceleration of particle be f Then in simple harmonic motion, acceleration f α –x                                        

 f = – ω2x, where ω2 is a constant

 

:. The restoring force acting on the particle is

 

F = mass x acceleration = mf = – m ω2x

 

Here the negative sign only gives the direction of force which is opposite to the direction displacement. Obviously, the magnitude of force is proportional to the displacement.

 

This Work done against the restoring force in displacing the particle from x to (x + dx) is   dW = m ω2x dx.

 

This work done is -stored in the particle in. the form of potential energy. Hence total work done in displacing the particle from mean position x = 0 to displaced position x = potential energy of the particle at displacement x



 

Kinetic Energy

 

From equation (2) of article 1.3, at an instant t, when the displacement of particle is x, the velocity of particle is



Kinetic energy of particle K = 1/2 mu2

 

K =1/2  m w2(a2 – x2)