find the expression for kinetic energy potential energy and Total energy of a particle executing SHM
Answers
In a simple harmonic oscillator, the energy oscillates between kinetic energy of the mass K = 12mv2 and potential energy U = 12kx2 stored in the spring. In the SHM of the mass and spring system, there are no dissipative forces, so the total energy is the sum of the potential energy and kinetic energy.
As an object falls from rest, its gravitational potential energy is converted to kinetic energy. Conservation of energy as a tool permits the calculation of the velocity just before it hits the surface. K.E. = J, which is of course equal to its initial potential energy.
A particle executes SHM with an amplitude 4 cm. At what displacement from the mean position its energy is half kinetic and half potential. when Kinetic energy is maximum potential energy is zero. when kinetic energy is half of its maximum, potential energy will become other half and both are equal.
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Derivation for the equation of Potenstial Energy: Let the work done on the object against gravity = W Work done, W = force × displacement Work done, W = mg × h Work done, W = mgh Since workdone on the object is equal to mgh, an energy equal to mgh units is gained by the object . This is the potential energy (Ep) of the object. Ep = mgh Derivation for the equation of Kinetic Energy: The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is v2 - u2 = 2aS This gives S = v 2 - u 2 2a We know F = ma. Thus using above equations, we can write the workdone by the force, F as W = ma × v 2 - u 2 2a or W = 1 2 m( v 2 - u 2 ) If object is starting from its stationary position, that is, u = 0, then W = 1 2 m v 2 It is clear that the work done is equal to the change in the kinetic energy of an object. If u = 0, the work done will be W = 1 2 m v 2 . Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek = ½ mv2.