Energy of Particle Performing Linear S.H.M.

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Kintetic Energy of Particle Performing Linear S.H.M.:

  • Consider a particle of mass ‘m’ which is performing linear S.H.M. of amplitude ‘a’ along straight line AB, with the centre O.  Let the position of the particle at some instant be at C, at a distance x from O.

Energy of SHM 01

This is an expression for the kinetic energy of particle S.H.M.

Thus the kinetic energy of the particle performing linear S.H.M. and at a dis the ance of x1 from mean position is given by

Energy of SHM 02



Special cases:

  • Case 1: Mean Position:

Kinetic energy of particle performing S.H.M. is given by

Energy of SHM 02

For mean position  x1 = 0



Energy of SHM 03

  • Case 2: Extreme position:

Kinetic energy of particle performing S.H.M. is given by

Energy of SHM 02

For mean position  x1 = a

Energy of SHM 04





Potential Energy of Particle Performing Linear S.H.M.:

  • Consider a particle of mass ‘m’ which is performing linear S.H.M. of amplitude ‘a’ along straight line AB, with the centre O.  Let the position of the particle at some instant be at C, at a distance x from O.

Energy of SHM 05

Particle at C is acted upon by restoring force which is given by F =  – mω²x

The negative sign indicates that force is restoring force.

  • Let. External force F’ which is equal in magnitude and opposite to restoring force acts on the particle due to which the particle moves away from the mean position by small distance ‘dx’ as shown. Then

F’ = mω²x



Then the work done by force F’ is given by

dW =  F’ . dx

dW = mω²x dx

The work done in moving the particle from position ‘O’ to ‘C’ can be calculated by integrating the above equation

Energy of SHM 06



This work will be stored in the particle as potential energy

Energy of SHM 07

This is an expression for the potential energy of particle performing S.H.M.



Special cases:

  • Case 1: Mean Position:

Potential energy of particle performing S.H.M. is given by

Energy of SHM 07

For mean position x1 = 0

∴  EP = 0

  • Case 2: Extreme position:

Potential energy of particle performing S.H.M. is given by

Energy of SHM 07



For mean position x1 = a

Energy of SHM 08

Total Energy of Particle Performing Linear S.H.M.:

  • The Kinetic energy of particle performing S.H.M. at a displacement of x1  from mean position is given by

Energy of SHM 02

  • The potential energy of particle performing S.H.M. at a displacement of x1  from mean position is given by

Energy of SHM 07



  • The total energy of particle performing S.H.M. at a displacement of x1  from the mean position is given by

Energy of SHM 09

  • Since for a given S.H.M., the mass of body m, angular speed ω and amplitude a are constant, Hence the total energy of a particle performing S.H.M. at C is constant. i.e. the total energy of a linear harmonic oscillator is conserved. It is same at all positions. The total energy of a linear harmonic oscillator is directly proportional to the square of its amplitude.

Variation of Kinetic Energy and Potential Energy in S.H.M Graphically.

Energy of SHM 10

Relation Between the Total Energy of particle and Frequency of S.H.M.: 

 

Energy of SHM 11

  •  The quantities in the bracket are constant. Therefore, the total energy of a linear harmonic oscillator is directly proportional to the square of its frequency.





Relation Between the Total Energy and Period of S.H.M.: 

 

Energy of SHM 12

  • The quantities in the bracket are constant. Therefore, the total energy of a linear harmonic oscillator is inversely proportional to the square of its period.

Expressions for Potential Energy, Kinetic Energy and Total Energy of a Particle Performing S.H.M. in Terms of Force Constant:

Energy of SHM 13

  • Potential energy: Potential energy of particle performing S.H.M. is given by

Energy of SHM 14

This is an expression for the potential energy of particle performing S.H.M. in terms of force constant.

  • Kinetic energy: Kinetic energy of particle performing S.H.M. is given by

Energy of SHM 15

This is an expression for Kinetic energy of particle performing S.H.M. in terms of force constant.

  • Total energy: Total energy of particle performing S.H.M. is given by

Total Energy of Particle

This is an expression for the total energy of particle performing S.H.M. in terms of force constant.

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