# A particle is executing SHM of amplitude r. AT a distance s from the mean position, the particle receives a blow in hte direction opposite to motion which instantaneously doubles the velocity. find the new amplitude.

yours katarnak Suresh
43 Points
11 years ago

Simple Harmonic Motion: A body is said to be in simple harmonic motion such that, at
any point its acceleration is directed always towards the mean position and it is directly
proportional to its displacement from the mean position.
Consider a particle moving along a circle of radius A with uniform angular velocity ω . Let
O be the centre of the circle and XX 1 and YY 1 are two mutually perpendicular diameters of
the circle as shown in figure. Let the particle is at P at any instant of time t Let PN be the
normal drawn on to the diameter YY 1 from P. As P moves on the circumference of the
circle, N moves on the diameter YY 1 . Let the angular displacement of the particle P is
∠XOP = θ
The displacement of N with respect to the fixed point O in the path is given by
ON = Y = A sin θ (or) Y = A sin ω t (Q θ = ω t ) ................. (1)
The acceleration of N ( aN ) is equal to the component of the centripetal acceleration of P
parallel to the diameter YY 1 .
The centripetal acceleration of P along PO is given by
a p = Aω 2 .............(2)
The centripetal acceleration parallel to NO.
aN = aP sin θ .............(3)
From equations (2) and (3) , we get,
aN = Aω 2 sin ωt
Using equation (1) ,
aN = −ω 2 y
The negative sign indicates that aN and Y are in opposite directions. Since ‘ ω ’ is a
constant then we can write aN α − y
i.e, the acceleration of N is directly proportional to the displacement in magnitude but in
opposite direction, always towards the fixed point ‘O’ in the path.
Hence the projection of uniform circular motion on t the diameter is simple harmonic.
(b) A = 0.04 m ; v = 50Hz ;
φ =π /3

y = A sin (ω t + φ ) = A sin ( 2π vt + φ )
y = 0.04 sin (100π t + π / 3 ) m.

MAhima Agarwal
24 Points
7 years ago
before the blow, we know the total energy of the system was 1/2 k r^2 since when the particle is at maximum amplitude, all the mechanical is in the form of elastic PE which is equal to 1/2 k x^2 where x is the displacement from equilibrium

just before the blow, the PE of the particle was 1/2 k s^2, so its KE at the time was

1/2 k r^2 - 1/2 k s^2 = 1/2 k(r^2-s^2)

if the blow doubled the velocity, the KE quadrupled (since KE depends on v^2), so the KE immediately after the blow was 2 k(r^2 - s^2)

now, the PE is still 1/2 k s^2, so the total energy of the system was

total energy = total PE + total KE = 1/2 k s^2 + 2 k(r^2 - s^2) = 2 k r^2 - 3/2 k s^2

since this is the total energy, it must be equal to the new total energy when the particles is at the farthest point, which we call X. The PE at X is

1/2 k X^2 and this equals total energy

so 1/2 k X^2 = 2 k r^2 - 3/2 k s^2

divide through by 1/2 k and obtain

X^2 = 4 r^2 - 3 s^2

and X = Sqrt[4 r^2 - 3s^2]