A simple harmonic oscillator is characterized by = cos . Find out the displacement at
which its kinetic energy is equal to its potential energy.
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Physics > Oscillations > Energy in Simple Harmonic Motion
Oscillations
Energy in Simple Harmonic Motion
Each and every object possesses energy, either while moving or at rest. In the simple harmonic motion, the object moves to and fro along the same path. Do you think an object possesses energy while travelling the same path again and again? Yes, it is energy in simple harmonic motion. Let’s learn how to calculate this energy and understand its properties.
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Questions
The potential energy function for a particle executing linear simple harmonic motion is given by U(_{x})= \frac{1}{2} \ kx^{2}U(
x
)=
2
1
kx
2
, where kk is the force constant. For k= 0.5 N m^{-1}k=0.5N m
−1
,the graph of U(x)U(x) versus xx is shown in figure. Show that a particle of total energy 1\ J1 J moving under this potential 'turns \ back'
′
turns back
′
when it reaches x= \pm \ 2mx=± 2m.
1 Verified answer
A particle free to move along the x-axis has potential energy given by U(x)=k[1-exp\left ( -x^{2} \right )]U(x)=k[1−exp(−x
2
)] for-\infty \leq x\leq +\infty ,−∞≤x≤+∞, where k is a positive constant of appropriate dimensions. Then:
1 Verified answer
An object performing S.H.M. with mass of 0.5\ kg0.5 kg, force constant 10N/m10N/m and amplitude 3 cm3cm . What is the speed at x=2 ?
1 Verified answer
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Energy in Simple Harmonic Motion
The total energy that a particle possesses while performing simple harmonic motion is energy in simple harmonic motion. Take a pendulum for example. When it is at its mean position, it is at rest. When it moves towards its extreme position, it is in motion and as soon as it reaches its extreme position, it comes to rest again. Therefore, in order to calculate the energy in simple harmonic motion, we need to calculate the kinetic and potential energy that the particle possesses.
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Browse more Topics under Oscillations
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Some Systems executing Simple Harmonic Motion
Energy in Simple Harmonic Motion
Periodic and Oscillatory Motion
Kinetic Energy (K.E.) in S.H.M
Kinetic energy is the energy possessed by an object when it is in motion. Let’s learn how to calculate the kinetic energy of an object. Consider a particle with mass m performing simple harmonic motion along a path AB. Let O be its mean position. Therefore, OA = OB = a.
The instantaneous velocity of the particle performing S.H.M. at a distance x from the mean position is given by
v= ±ω √a2 – x2
∴ v2 = ω2 ( a2 – x2)
∴ Kinetic energy= 1/2 mv2 = 1/2 m ω2 ( a2 – x2)
As, k/m = ω2
∴ k = m ω2
Kinetic energy= 1/2 k ( a2 – x2) . The equations Ia and Ib can both be used for calculating the kinetic energy of the particle.