Oscillations

Question 1

What is periodic motion?

Answer 1

Any motion that repeats itself over and over again at regular intervals of time

Question 2

Oscillatory Motion / Harmonic

Answer 2

Body moves back and forth repeatedly about its mean position

Question 3

Fourier theorem

Answer 3

Any arbitrary function F(t) with period T can be expressed as the unique combination of sine and cosine functions fn (t) and gn (t) with suitable coefficients

Question 4

Harmonic Functions

Answer 4

Periodic functions that can be represented by a sine or cosine curve

Question 5

Non-harmonic Functions

Answer 5

Periodic functions which cannot be represented by single sine or cosine function

Question 6

Simple Harmonic Motion

Answer 6

If it moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from the mean position and is always directed towards the mean position

Question 7

Prove to find the differential equation of SHM

Answer 7

-
d^x / dt^2 + (omega)^2 x = 0

Question 8

Find the of SHM:
1. Displacement
2. Time period

Answer 8

1. x = A cos(omega t + psi not)
2. T = 2pi (root m / k)

Question 9

When to use x = A sin omega t and x = A cos omega t

Answer 9

x = A cos omega t –> When it starts from maximum displacement
x = A sin omega t –> When it starts from equilibrium position

Question 10

Value of Spring Constant

Answer 10

k = m omega^ 2 = F / x

Question 11

Harmonic Oscillator

Answer 11

A particle executing simple harmonic motion

Question 12

Amplitude

Answer 12

Maximum displacement of the osciallating particle on either side of its mean position

Question 13

Oscillation

Answer 13

One complete back and forth motion of a particle starting and ending at the same point

Question 14

Angular frequency

Answer 14

Quantity obtained by multiplying frequency by a factor of 2pi

Question 15

Phase

Answer 15

Gives the state of the particle as regards its position and the direction of motion at that instant

Question 16

Initial phase / Epoch

Answer 16

Phase of a vibrating particle corresponding to time t = 0

Question 17

Circle of Reference

Answer 17

A circle of reference can refer to a circle used to establish the center of a component or to a circle used to describe the motion of a particle undergoing simple harmonic motion (SHM)

Question 18

Displacement formula from centre of circle

Answer 18

x = A cos (omega t + psi not)
A = amplitude
x = Displacement
omega t = angular frequency
psi not = initial phase

Question 19

Acceleration in SHM

Answer 19

a(t) = - omega ^2 A cos (omega t + psi not) = - omega ^2 x

Question 20

Acceleration amplitude

Answer 20

Acceleration amplitude is the maximum acceleration of an oscillating particle

Question 21

Acceleration amplitude formula

Answer 21

a max = omega^2 A = (2pi / T)^2 A

Question 22

acceleration in SHM

Answer 22

a(t) = omega ^2 A cos (omega t + pi)

Question 23

Kinetic Energy in SHM

Answer 23

KE = 1/2 m omega^2 A^2 sin^2 (omega t + psi not) = 1/2 m omega^2 (A^2 - x^2)

Question 24

Potential Energy in SHM

Answer 24

U = 1/2 kx^2 = 1/2 m omega^2 x^2 = 1/2 m omega^2 A^2 cos^2 (omega t + psi not)

Question 25

Total Energy in SHM

Answer 25

E = 1/2 k A^2 = 1/2 m omega^2 A^2 = 2 pi^2 m v^2 A^2

Question 26

Time period in SHM

Answer 26

T = 2pi root (m / k)

Question 27

Oscillation frequency for springs connected in parallel

Answer 27

mu = 1/2pi [root (k1 + k2) / m]

Question 28

Oscillation frequency for springs connected in series

Answer 28

mu = 1 / 2pi [root (k1k2 / m (k1 + k2))]

Question 29

Simple Pendulum

Answer 29

Consists of a point-mass suspended by a flexible, inelastic and weightless string from a rigid support of infinite mass

Question 30

Time period of a bob of simple pendulum

Answer 30

T = 2pi root (l / g)

Question 31

Free Oscillations

Answer 31

If a body, capable of oscillation, is slightly displaced from its position of equiliibrium and left to itself, it starts oscillating with a frequnecy of its own

Question 32

Damped Oscillations

Answer 32

Oscillations in which the amplitude decreases gradually with the passage of time