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Practice each question, then open the answer or worked solution.
Find the displacement of a simple harmonic oscillator at which its PE is half of the maximum energy of the oscillator.
The mean position of the oscillator at x with PE = 1/2 kx2
PE = -m2x2
When PE is maximum, KE = 0 at x = A
E = 1/2 mω2A2
PE = 1/2 total energy
x2 = 1/2 A2
x = ± A/√2
A body of mass m is situated in a potential field U(x) = U0 (1 – cos αx) when U0 and α are constants. Find the time period of small oscillations.
The time period of the small oscillation is T = 2π/α √m/U0
A mass of 2 kg is attached to the spring of spring constant 50 N/m. The block is pulled to a distance of 5 cm from its equilibrium position at x = 0 on a horizontal frictionless surface from rest at t = 0. Write the expression for its displacement at anytime t.
x = 5 sin 5t
Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme position making an angle of 2o to the right with the vertical, the other pendulum makes an angle of 1o to the left of the vertical. What is the phase difference between the pendulums?
θ1 = θ0 sin (wt + δ1)
θ2 = θ0 sin (wt + δ2)
for first, θ = 2o
for second, θ = -1o
δ1 – δ2 = 120o
Long AnswersA person normally weighing 50 kg stands on a massless platform which oscillates up and down harmonically at a frequency of 2 s-1 and an amplitude 5 cm. A weighing machine on the platform gives the person’s weight against time.
a) will there be any change in the weight of the body during the oscillation?
b) if the answer to part a) is yes, what will be the maximum and minimum reading in the machine and at which position?
a) The change in the weight of the body during the oscillation is 885 N
b) The maximum weight is 885 N, while the minimum weight is 95.5 N.
A body of mass m is attached to one end of a massless spring which is suspended vertically from a fixed point. The mass is held in hand so that the spring is neither stretched nor compressed. Suddenly the support of the hand is removed. The lowest position attained by the mass during oscillation is 4 cm below the point, where it was held in hand.
a) what is the amplitude of oscillation?
b) find the frequency of oscillation?
The PE of spring = gravitational PE lost by the man
PE = mgx
1/2 kx2 is the PE due to spring
x = 2mg/k
a) the amplitude of the spring is 2cm
b) frequency of oscillation is v = 3.52 Hz
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute SHM with a time period where m is the mass of the body and ρ is the density of the liquid.
The SHM executed by the cylindrical wood log is given as
T = 2π√m/Aρg
One end of a V-tube containing mercury is connected to a suction pump, and the other end to the atmosphere. The two arms of the tube are inclined horizontally at an angle of 45o each. A small pressure difference is created between two columns when the suction pump is removed. Will the column of mercury in V-tube execute simple harmonic motion? Neglect capillary and viscous forces. Find the time period of oscillation.
The initial height of the columns is h0
Let dx be the element that needs to be considered
The mass is dm = A.dx.ρ
PE on the left of dm = (dm)gh
Total PE = A ρg h12/2
When the above equation is differentiated along with KE where KE is A ρlv2
We get the equation of SHM oscillator along with the standard equation of SHM when the total energy is obtained by adding PE and KE
T = 2π√l/g
A tunnel is dug through the centre of the earth. Show that a body of mass ‘m’, when dropped from rest from one end of the tunnel, will execute the simple harmonic motion.
Acceleration due to gravity of the earth inside the earth is g’
g’ = g(1-d/R)
R-d = y
g’ = g y/R
F = -mg y/R
F is proportional to (-y)
When the body is in a tunnel, the motion is SHM. ma = -mg’
T = 2π √R/g
A simple pendulum of time period 1s and length l is hung from a fixed support at O, such that the bob is at a distance H vertically above A on the ground. The amplitude is θo. The string snaps at θ = θ0/2. Find the time taken by the bob to hit the ground. Also, find the distance from A where bob hits the ground. Assume θ0 to be small so that sin θo, θo and cos θ0 1.
At t = t1 and θ= θ0/a
T = 1 sec
θ0/2 = θ0 cos 2π/T t1
d θ/dt = – θ0 2π sin 2πt
At t = 1/6, that is when θ = θ0/2
d θ/dt = – θ0π√3
v/l = – θ0π√3
v = -θ0π√3l
The distance between the bob from A is θ0l [1/2 – π√6H/g]
Oscillation has many different uses and real-life applications, as the examples such as: