The total energy (E) of an oscillating particle is equal to the sum of its kinetic energy and potential energy if conservative force acts on it. The velocity of a particle executing SHM at a position where its displacement is y from its mean position is v = ω rt( a2-y2)

**Energy in simple harmonic motion **

The total energy
(E) of an oscillating particle is equal to the sum of its kinetic energy and
potential energy if conservative force acts on it. The velocity of a particle
executing SHM at a position where its displacement is y from its mean position
is v = ω rt( a^{2}-y^{2})

**Kinetic energy **

Kinetic energy
of the particle of mass m is

K = ? n [ω rt( a^{2}-y^{2})
]

K = ? m [ω^{2}
( a^{2}-y^{2}) ]
?.(1)

**Potential energy **

From definition
of SHM F = ?ky the work done by the force during the small displacement dy is
dW = −F.dy = −(−ky) dy = ky dy

∴
Total work done for the displacement y is,

W = ∫ dW = ∫_{0}^{y} ky dy

W = ∫_{0}^{y} mω^{2} y dy

∴W
= 1/ 2 m ω^{2} y^{2}

This work done
is stored in the body as potential energy

U = 1 /2 m ω^{2}
y^{2}

Total energy E =
K + U

= 1/ 2 mω^{2}
(a^{2} − y^{2}) + 1/ 2 m ω^{2} y^{2}

= 1/ 2 m ω^{2}
a^{2}

Thus we find that the total energy of a
particle executing simple harmonic motion is 1 /2 m ω^{2} a^{2}

**Special cases **

(i)When the
particle is at the mean position y = 0, from eqn (1) it is known that kinetic
energy is maximum and from eqn. (2) it is known that potential energy is zero.
Hence the total energy is wholly kinetic

E=K^{max} = 1/ 2 mω^{2}a^{2}

(ii) When the
particle is at the extreme position y = +a, from eqn. (1) it is known that
kinetic energy is zero and from eqn. (2) it is known that Potential energy is
maximum. Hence the total energy is wholly potential.

E = U^{max}
= ? m ω^{2} a^{2}

(iii)when y =
a/2 ,

K = 1 /2 m ω^{2}
[a^{2} - a^{2}/4]

∴K
= ?(1/2mω^{2} a^{ 2})

K = 3 /4 E

U = 1/2 mω^{2}
(a^{ 2}/2) = ?(1/2 mω^{2} a^{ 2})

U = 1/4E

If the
displacement is half of the amplitude, K = 3/ 4 E and U = 1 /4 E. K and U are
in the ratio 3 : 1,

E=K+U=1/2 mω^{2}
a^{ 2}

At any other
position the energy is partly kinetic and partly potential.

This shows that
the particle executing SHM obeys the law of conservation of energy.

**Graphical representation of energy **

The values of K
and U in terms of E for different values of y are given in the Table 6.2. The
variation of energy of an oscillating particle with the displacement can be
represented in a graph as shown in the Fig..

**Energy of SHM**

Y : 0 a/2 , a , -a/2 , -a

Kinetic energy :
E , ? E , 0 ? E, 0

Potential energy
: 0 , 1/4E, E , 1/4E , E

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