VELOCITY AND ACCELERATION IN SIMPLE HARMONIC MOTION -

we know that the equation of displacement executing S.H.M is

$\fn_cm \large \left [ y=A\sin(\omega t+\phi) \right ]\;\;\;-----(1)$

If we differentiate y with respect to t, then its velocity becomes

$\fn_cm \large v=\frac{\mathrm{d} y}{\mathrm{d} t}$

$\fn_cm \large v=\frac{\mathrm{d} [A\sin(\omega t+\phi)]}{\mathrm{d} t}$

$\fn_cm \large \left [ v=A\omega\cos(\omega t+\phi) \right ]\;\;\;-----(2)$

It is also executed in S.H.M with the same time period

From geometry, it can be view easily. The speed of a particle $\fn_cm \large v$ in uniform circular motion is its angular speed $\fn_cm \large \omega$ times the radius of the circle. i.e $\fn_cm \large v=A\omega$

Its direction is always tangent at that point.

It is clear from figure, the velocity of projection particle on Y-axis at time t is

$\fn_cm \large v=A\omega\cos(\omega t+\phi)$

If we differentiate again w.r.t time, then its acceleration becomes.

$\fn_cm \large a=\frac{\mathrm{d} v}{\mathrm{d} t}$

$\fn_cm \large a=\frac{\mathrm{d} [A\omega\cos(\omega t+\phi)]}{\mathrm{d} t}$

$\fn_cm \large \left [ a=-A\omega^2\sin(\omega t+\phi) \right ]\;\;\;-----(3)$

and its direction is always towards the mean position.

From geometry, when the particle moves in a uniform circular motion of angular speed $\fn_cm \large \omega$ with radius A. Then its centripetal acceleration at any point is $\fn_cm \large a\omega ^2$ which is always towards the centre.

It is clear from fig, the acceleration of projection particle on Y-axis at time t is

$\fn_cm \large a=-A\omega^2\sin(\omega t+\phi)$

Where -ve sign show that it is opposite direction to the +ve direction of the Y-axis.

from (1) and (3)

$\fn_cm \large \mathbf{\left [ a=-\omega^2 y \right ]}$

from (1)

$\fn_cm \large \sin(\omega t+\phi)=\frac{y}{A}$

and from (2)

$\fn_cm \large \cos(\omega t+\phi)=\frac{v}{A\omega}$

Here squaring both sides and added, we get

$\fn_cm \large \sin^2(\omega t+\phi)+\cos^2(\omega t+\phi)=\frac{y^2}{A^2}+\frac{v^2}{A^2 \omega^2}$

$\fn_cm \large \Rightarrow \frac{y^2\omega^2 + v^2}{A^2 \omega^2}=1$

$\fn_cm \large \Rightarrow y^2\omega^2+v^2=A^2\omega^2$

$\fn_cm \large \Rightarrow v^2=A^2\omega^2-y^2\omega^2$

$\fn_cm \large \left [ v=\omega\sqrt{A^2-y^2} \right ]$

a. A body oscillates with SHM according to the equation (in SI units), x = 5 cos [2π t + π/4].At t = 1.5 s, calculate the (a) displacement,(b) speed and (c) acceleration of the body.

b. A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is
(a) at the end A,
(b) at the end B,
(c) at the mid-point of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.

c. Which of the following relationships between the acceleration a and the displacement x of a particle involve simple harmonic motion?
(a) a = 0.7x
(b) a = –200 $\fn_cm x^2$
(c) a = –10x
(d) a = 100 $\fn_cm x^3$

d. The motion of a particle executing simple harmonic motion is described by the displacement function, x(t) = A cos (ωt + φ ). If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

e. A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.

f. A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]