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,
Fnet=md2xdt2=−kxwhere k is the spring constant.
Solving for x gives:
x(t)=c1cos(ωt)+c2sin(ωt)=Acos(ωt−φ)where
ω=√km A=√c21+c22 tanφ=c2c1Also, the velocity is given by:
v(t)=dxdt=−Aωsin(ωt−φ) Speed=ω√A2−x2maximum speed=ωA
This maximum speed is attained at the equilibrium point.
Also, the acceleration is given by:
a(t)=d2xdt2=−Aω2cos(ωt−φ)=−ω2xmaximum acceleration = Aω2
The maximum acceleration is attained at extreme points.
From above and Hooke’s Law, we get
a(x)=−ω2x=−kxmwhich implies that
ω2=kmwhere
ω=2πf=2πTEnergy: Kinetic Energy,
K(t)=12mv2(t)=12mω2A2sin2(ωt−φ)=12kA2sin2(ωt−φ)Potential Energy,
U(t)=12kx2(t)=12kA2cos2(ωt−φ)and
Total Energy,
E=K+U=12kA2PYQs
Simple Harmonic Motion
1) A particle moving along the y−axis has an acceleration Fy 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
2π√F1<T<2π√F2where F1 and F2 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∘ on either side of the vertical line, T lies between 2π√l/g and 2π√l/g√π/3.
[2019, 20M]
2) A particle moving with simple harmonic motion in a straight line has velocities v1 and v2 at the distances x1 and x2 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 23a, 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 Tπcos′(ba).
[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 mg 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−tan12)√21g.
[2009, 20M]
7) A particle is performing simple harmonic motion of period T about a centre O. It passes through a point P(OP=p) with velocity v in the direction OP. Show that the time which elapses before it returns to P is Tπtan′vT2πp.
[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]