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Obtain the expressions of kinetic energy potential energy and total energy in simple harmonic motion. |
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Answer» Solution :Kinetic energy : DISPLACEMENT of SHM particle at any instant `x= A COS (omega t +phi)` where A = amplitude, `omega`= angular frequency velocity `v= (d(x))/(dt)= (d)/(dt)[ A cos (omega t+ phi)]` `therefore v= -A omega sin (omega t+phi)` Now kinetic energy, `K= (1)/(2) mv^(2)` `=(1)/(2) mA^(2) omega^(2) sin^(2) (omega t+phi)` `therefore K= (1)/(2) kA^(2) sin^(2) (omega t+phi)"""......"(1) [therefore m omega^(2) = k]` Hence, it is a periodic function of time, being zero when the displacement is maximuml and maximuml when the particle is at the mean position. The period of kinetic energy is `(T)/(2)` Potential energy : Potential energy is possible only for conservative forces and restoring force is the conservative force in SHM. If the restoring force exerted on a particle of SHM from its mean position is x then `F= -KX` The work opposite to restoring force for small displacement dx of a particle is `dW= -F dx` `therefore dW = -kx dx` Now work done for displacement of particle from `x=0` to x=x, `W= int dW` `= int_(0)^(x) -kx dx` `= -k[(x^2)/(2)]_(0)^(x)` `= -(kx^2)/(2)` `= -(1)/(2) kx^(2)` The work done opposite to restoring force is stored as potential energy in particle. Potential energy in displacement x of particle is `U = -W` `=(1)/(2) kx^(2)` `= (1)/(2)m omega^(2) x^(2)` `=(1)/(2) m omega^(2) A^(2) cos^(2) (omega t+ phi)` `therefore U= (1)/(2) kA^(2) cos^(2) (omega t +phi)` Hence, the potential energy of a particle executing simple harmonic motion is also periodic. At mean position potential energy is zero and at extreme points it is maximum. Hence, the period of potential energy is `(T)/(2)`. Total energy : The sum of kinetic and potential energy of SHM is knownf as total energy E. `therefore E= K+U` `therefore E= (1)/(2) kA^(2) sin^(2)( omega t+phi) +(1)/(2) kA^(2) cos^(2) (omega t+phi)` `therefore E= (1)/(2) kA^(2) [sin^(2) (omega t+phi)+ cos^(2) (omega t+phi)]` `=(1)/(2) kA^(2) [therefore sin^(2) (omega t+phi)+ cos^(2) (omega t+phi)=1]` or `E= (1)/(2) m omega^(2) A^(2)` Hence, `E propto m, E propto omega^(2)" and "E propto A^(2)` Thus, mechanical energy is proportional to mass and square of angular frequency and varies directly to the square of amplitude but mechnical energy does not depend on the displacement and time. |
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