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Physics 1 (Assignment 1)

1 A weight suspended from a spring extends it by x = 9.8cm. What will be theoscillation period of the weight when it is pushed gently in the vertical direction? The

logarithmic decrement is equal to = 3.1[ans: T = 0.70sec]

2 Find the quality factor of an oscillator whose displacement amplitude decreases by

a factor = 2.0 every n = 110 oscillations.[ans: Q = 500]

3 A particle is displaced from the equilibrium position by a distance l = 1.0cm andthen left alone. What is the distance that the particle covers till its oscillations die down,

if the logarithmic decrement is equal to = 0.020?[ans: s = 2m nearly]

4 Find the quality factor of a simple pendulum of length l = 50 cm if during the time

interval = 5.2minutes its total mechanical energy decreases by a factor = 4.0104.[ans: Q = 1.3

102]

5 A spring mass system has an undamped time period T0 = 2seconds. It is thensubject to critical damping. The mass is pulled to one side and then released from rest

at t= 0. Find the time in seconds, (0 < < ), at which the damping force exactlybalances the spring force.

[ans:= 1second]

6 A particle of mass m can perform undamped harmonic oscillations about a point

x = 0, with a natural frequency 0. At the moment t = 0, when the particle is inequilibrium, a force Fx = F0cost along the x axis is applied to it. Find the law offorced oscillations x(t) of the particle.

[ans: x =

F0

m

2 20(cos0t cost)]

7 A particle of mass m can perform undamped harmonic oscillations due to an elastic

restoring force with a spring constant k. When the particle is in equilibrium, a constant

force F is applied to it for seconds. Find the amplitude of oscillation of the particleafter the force ceases.

[ans: a = (2F

k) | sin(0/2) |, where 0 =

k

m]

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8 A forced harmonic oscillator has equal displacement amplitude at the frequencies

1 = 400s1

and 2 = 600s1

. Find the resonance frequency at which the displacementamplitude is maximum.

[ans: res = 5.1102s1]

9 The velocity amplitude of a particle is equal to half its maximum value at frequen-

cies 1and 2 of an external harmonic force. Find:(a) The frequency corresponding to the velocity resonance;

(b) The damping coefficient and the damped oscillation frequency of the parti-cle.

[ans: (a) 0 =12; (b) =| 12 | /2

3, =

12 [(12)2/12]]

10 A mass m = 50g is suspended by a massless spring with a spring constant =

20.0N/m. The mass performs steady state vertical oscillations of amplitude a = 1.3cmdue to an external harmonic force of frequency = 25.0s1. The displacement lagsbehind the force by an angle = (3/4). Find:

(a) the quality factor of the given oscillator;

(b) the work performed by the external force in one oscillation.

[ans: (a) Q = 2.2 (b) W= 6mJ]

11 A mass m suspended by a massless spring performs vertical oscillations with a

damping coefficient . Its natural frequency is 0. The mass performs vertical os-cillations in the steady state due to an external vertical harmonic force varying as

F= F0cost. Find:(a) the mean power < P> developed by the force in one cycle;(b) the frequency at which < P > is maximum and the value of its maximum

average power < P>max.

[ans: (a) < P>=F20

2/m

(20 2)2 + 422; (b) = 0, < P>max=

F204m

]

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