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Simple Harmonic Motion

The equation of motion for one-dimensional simple harmonic motion can be obtained by means of Newton’s Second Law of Motion and Hooke’s Law. Mathematically,

F_{n e t} = m \frac{d^{2} x}{d t^{2}} = - k x

where $k$ is the spring constant.

Solving for $x$ gives:

x (t) = c_{1} \cos (ω t) + c_{2} \sin (ω t) = A \cos (ω t - φ)

where

ω = \sqrt{\frac{k}{m}}

A = \sqrt{c_{1}^{2} + c_{2}^{2}}

\tan φ = \frac{c_{2}}{c_{1}}

Also, the velocity is given by:

v (t) = \frac{d x}{d t} = - A ω \sin (ω t - φ)

S p e e d = ω \sqrt{A^{2} - x^{2}}

maximum speed= $ω A$

This maximum speed is attained at the equilibrium point.

Also, the acceleration is given by:

a (t) = \frac{d^{2} x}{d t^{2}} = - A ω^{2} \cos (ω t - φ) = - ω^{2} x

maximum acceleration = $A ω^{2}$

The maximum acceleration is attained at extreme points.

From above and Hooke’s Law, we get

a (x) = - ω^{2} x = \frac{- k x}{m}

which implies that

ω^{2} = \frac{k}{m}

where

ω = 2 π f = \frac{2 π}{T}

Energy: Kinetic Energy,

K (t) = \frac{1}{2} m v^{2} (t) = \frac{1}{2} m ω^{2} A^{2} \sin^{2} (ω t - φ) = \frac{1}{2} k A^{2} \sin^{2} (ω t - φ)

Potential Energy,

U (t) = \frac{1}{2} k x^{2} (t) = \frac{1}{2} k A^{2} \cos^{2} (ω t - φ)

and

Total Energy,

E = K + U = \frac{1}{2} k A^{2}

PYQs

Simple Harmonic Motion

1) A particle moving along the $y - a x i s$ has an acceleration $F_{y}$ towards the origin, where $F$ is a psoitive and even function of $y$ . The periodic time, when the particle vibrates between $y = - a$ and $y = a$ , is $T$ . Show that

\frac{2 π}{\sqrt{F_{1}}} < T < \frac{2 π}{\sqrt{F_{2}}}

where $F_{1}$ and $F_{2}$ are the greatest and the least values of $F$ within the range $[- a, a]$ . Further, show that when a simple pendulum of length $l$ oscillates through $30^{\circ}$ on either side of the vertical line, $T$ lies between $2 π \sqrt{l / g}$ and $2 π \sqrt{l / g} \sqrt{π / 3}$ .

[2019, 20M]

2) A particle moving with simple harmonic motion in a straight line has velocities $v_{1}$ and $v_{2}$ at the distances $x_{1}$ and $x_{2}$ respectively from the center of its path. Find the period of its motion.

[2018, 12M]

3) A body moving under SHM has an amplitude $^{'} a^{'}$ and time period $^{'} T^{'}$ . If the velocity is trebled, when the distance from mean position is $\frac{2}{3} a$ , the period being unaltered, find the new amplitude.

[2015, 10M]

4) A particle is performing a simple harmonic motion (SHM) of period $T$ about a centre $O$ with amplitude $a$ and it passes through a point $P$ , where $O P = b$ in the direction $O P$ . Prove that the time which elapses before it returns to $P$ is $\frac{T}{π} \cos^{'} (\frac{b}{a})$ .

[2014, 10M]

5) A body is performing SHM in a straight line $O P Q$ . Its velocity is zero at points $P$ and $Q$ whose distances from $O$ are $x$ and $y$ respectively and its velocity is $v$ at the mid-point between $P$ and $Q$ . Find the time of one complete oscillation.

[2013, 10M]

6) One end of a light elastic string of natural length $I$ and modulus of elasticity 2 $m g$ is attached to a fixed point $O$ and the other end to a particle of mass $m$ . The particle initially held at rest at $O$ is let fall. Find the greatest extension of the string during the motion and show that the particle will reach $O$ again after a time $(π + 2 - \tan^{1} 2) \sqrt{\frac{21}{g}}$ .

[2009, 20M]

7) A particle is performing simple harmonic motion of period $T$ about a centre $O$ . It passes through a point $P (O P = p)$ with velocity $v$ in the direction $O P$ . Show that the time which elapses before it returns to $P$ is $\frac{T}{π} \tan^{'} \frac{v T}{2 π p}$ .

[2007, 15M]

8) An elastic string of natural length $a + b$ , where $a > b$ , and modulus of elasticity $λ$ , has a particle of mass $m$ attached to it at a distance $a$ from one end which is fixed to a point $A$ of a smooth horizontal plane. The other end of the string is fixed to a point $B$ so that string is just unstretched. If the particle be held at $B$ and then released, find the periodic time and the distance in which the particle will oscillate to and fro.

[2003, 15M]

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