SHM
We will cover following topics
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,
\[F_{\mathrm{net}}=m \dfrac{\mathrm{d}^{2} x}{\mathrm{d} t^{2}}=-k x\]where \(k\) is the spring constant.
Solving for \(x\) gives:
\[x(t)=c_{1} \cos (\omega t)+c_{2} \sin (\omega t) = A \cos (\omega t-\varphi)\]where
\[\omega=\sqrt{\dfrac{k}{m}}\] \[A=\sqrt{c_{1}^{2}+c_{2}^{2}}\] \[\tan \varphi=\dfrac{c_{2}}{c_{1}}\]Also, the velocity is given by:
\[v(t)=\dfrac{\mathrm{d} x}{\mathrm{d} t}=-A \omega \sin (\omega t-\varphi)\] \[Speed = \omega \sqrt{A^{2}-x^{2}}\]maximum speed=\(\omega A\)
This maximum speed is attained at the equilibrium point.
Also, the acceleration is given by:
\[a(t)=\dfrac{\mathrm{d}^{2} x}{\mathrm{d} t^{2}}=-A \omega^{2} \cos (\omega t-\varphi)=-\omega^{2} x\]maximum acceleration = \(A \omega^{2}\)
The maximum acceleration is attained at extreme points.
From above and Hooke’s Law, we get
\[a(x)=-\omega^{2} x= \dfrac{-kx}{m}\]which implies that
\[\omega^{2}=\dfrac{k}{m}\]where
\[\omega=2 \pi f = \dfrac{2 \pi}{T}\]Energy: Kinetic Energy,
\[K(t)=\dfrac{1}{2} m v^{2}(t)=\dfrac{1}{2} m \omega^{2} A^{2} \sin ^{2}(\omega t-\varphi)=\dfrac{1}{2} k A^{2} \sin ^{2}(\omega t-\varphi)\]Potential Energy,
\[U(t)=\dfrac{1}{2} k x^{2}(t)=\dfrac{1}{2} k A^{2} \cos ^{2}(\omega t-\varphi)\]and
Total Energy,
\[E=K+U=\dfrac{1}{2} k A^{2}\]PYQs
Simple Harmonic Motion
1) A particle moving along the \(y-axis\) 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
\[\dfrac{2\pi}{\sqrt{F_1}}<T<\dfrac{2\pi}{\sqrt{F_2}}\]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 \(2\pi\sqrt{l/g}\) and \(2\pi\sqrt{l/g}\sqrt{\pi/3}\).
[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 \(\dfrac{2}{3} a\), 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 \(OP =b\) in the direction \(OP\). Prove that the time which elapses before it returns to \(P\) is \(\dfrac{T}{\pi} \cos ^{\prime}\left(\dfrac{b}{a}\right)\).
[2014, 10M]
5) A body is performing SHM in a straight line \(OPQ\). 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 \(\mathrm{mg}\) is attached to a fixed point \(\mathrm{O}\) and the other end to a particle of mass \(\mathrm{m}\). The particle initially held at rest at \(\mathrm{O}\) 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 \(\left(\pi+2-\tan ^{1} 2\right) \sqrt{\dfrac{21}{g}}\).
[2009, 20M]
7) A particle is performing simple harmonic motion of period \(T\) about a centre \(O\). It passes through a point \(\mathrm{P}(\mathrm{OP}=\mathrm{p})\) with velocity \(v\) in the direction \(OP\). Show that the time which elapses before it returns to \(\mathrm{P}\) is \(\dfrac{T}{\pi} \tan ^{\prime} \dfrac{v T}{2 \pi p}\).
[2007, 15M]
8) An elastic string of natural length \(a+b\), where \(a>b\), and modulus of elasticity \(\lambda\), 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]