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the total energy of the particles executing SHM and show grafically the variations of PE AND KE with time in SHM.What is the frequency of the engine with respect to the frequency of the particle executing SHM?

Ditsa Tomar , 7 Years ago
Grade 11
anser 1 Answers
Arun
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)
 
 
Total Energy
 
Total energy of particle = Potential energy + Kinetic energy
=1/2 mwx2 +1/2 mw2 (a2-x2)
= 1/2  m w^2a^2
 
But w = 2πn (where n is frequency)
[ E = 2π^2m  n^2a^2]
Last Activity: 7 Years ago
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