Oscillations

Question 1

Defn of oscillation

Answer 1

Is a special periodic motion in which the oscillator moves to and fro abt an eqm position

Question 2

Defn of simple harmonic motion

Answer 2

The motion of a particle abt a fixed pt such that its acc is prop to its displacement from the fixed pt and is always directed towards the pt

Question 3

Eqn for simple harmonic motion

Answer 3

a = - ω^2 x

Question 4

Defn of angular frequency

Answer 4

Rate of change of phase angle of the oscillation and is equal to the product of 2π and its frequency

Question 5

Eqn of angular freq

Answer 5

ω = dΘ / dt (rad s-1)
= 2π / T
= 2πf

Question 6

Defn of amplitude

Answer 6

Mag of the max displacement of the particle from its eqm position

Question 7

For a pendulum, where is the max/0 ACC & speed and where is acc directed at at each pt

Answer 7

Acc is always directed to O
Mag of ACC is 0 at O and max at A&B
Speed at O is max and 0 at A&B

Question 8

Eqn for angular freq for spring mass system

Answer 8

ω = √ (k/m) = 2πf

Question 9

Eqn for angular freq for simple pendulum system

Answer 9

ω = √ (g/L) = 2πf

Question 10

Eqn for instantaneous disp
when obj starts from eqm position
when obj starts from amp

Answer 10

X = Xo sin(ωt)
X = Xo cos(ωt)

Question 11

graph for 2 instantaneous disp

Answer 11

sin & cos graph with the max n min pts at +Xo & -Xo

Question 12

Eqn for velocity
from eqm
from amp

Answer 12

Just differentiate
X = ωXo cos(ωt)
X = -ωXo sin(ωt)

Question 13

Eqn for Vmax

Answer 13

±ωXo

Question 14

Eqn for acc
from eqm
from amp

Answer 14

a = - ω^2 Xo sin(ωt)
a = - ω^2 Xo cos(ωt)

Question 15

Draw V-Xo graph

Answer 15

check notes- circle shape

Question 16

Draw a - Xo graph

Answer 16

straight line graph with negative grad

grad = - ω^2

Question 17

Defn of Damping

Answer 17

damping is the process whereby energy is removed from an oscillating system

Question 18

defn of damped oscillation

Answer 18

is one where the total energy and amplitude of the system decreases exponentially with time due to energy losses through dissipative forces such as friction and air resistance

Question 19

Types of damping & defn of each

Answer 19

Light critical heavy
light - results in oscillation whereby the amp decays exponentially with time. The freq of oscillation is slightly smaller than the undamped freq
Critical - results in no oscillation and the system returns to the eqm position in the shortest time
heavy - results in no oscillation and the system takes a long time to return to its eqm position

Question 20

Types of damping on x-t graph

Answer 20

check notes

Question 21

Application: damping in car suspension system

Answer 21

should be critical damping
without suspension - spring will extend and release energy at an uncontrolled rate, bounce at natural freq till energy is used up
Heavy damping - still have a compressed spring by the time P is reached, cannot respond to the sudden drop

Question 22

Defn of forced oscillations

Answer 22

oscillations are maintained by an external driving force. The system oscillates at the freq of the driving force

Question 23

Defn of resonance

Answer 23

Occurs when a system responds at maximum amp to an external driving force. Thus occurs when the freq of the driving force is equal to the natural freq of the driven system, where there is maximum transfer of energy to the driven system.

Question 24

What does amp of forced oscillation depend (2)

Answer 24

1. damping of the system
   increased damping –> amp of resonance decrease –> resonance freq decreases as natural period of oscillating system increases
2. relative values of the driver freq f and the natural freq f0 of the system - how far f is from f0
   largest amp = f is approx equal to f0

Question 25

how does the amp-driver freq graph look like for natural, light, heavy damping

Answer 25

check notes

Question 26

What happens to the light damping graph when mass is increased

Answer 26

mass increases –> freq decreases –> inertia increases –> comp??? increases

Question 27

How does the expression (eqn) / graph show simple harmonic motion

Answer 27

check notes

Question 28

eqn for max vel

Answer 28

v = ωx0

Question 29

eqn for acc

Answer 29

a = - ω^2 x

Question 30

Eqn for max Ek/Pe/total energy

Answer 30

1/2 m ω^2 Xo^2

Question 31

Eqn for vel (in formulae list)

Answer 31

v = ±ω√(X0^2 - X^2)

Question 32

What happens to F-x (OR a-x) graph when there’s friction

Answer 32

Right side of the eqm pt (towards the right): Fnet = Fspring + f
Right side of the eqm pt (towards the left): Fnet = Fspring - f
…

So should be below/above the org straight line graph

some sort of spiral

Question 33

At which pt does obj leave the ‘platform’ - how to find?

Answer 33

It’s always above the eqm pt

mg - N = mω^2x
When N=0,
g = ω^2x

Displacement where it’ll loose contact = x = g/ω^2

Question 34

How to calc total spring constant for springs in series & parallel

Answer 34

Parallel: Knet = K1 + K2

Series: 1/Knet = 1/K1 + 1/K2

Question 35

How to convert a-t to x-t graph

Answer 35

literally flip it abt x-axis

Question 36

Wrt the exp provided, show that the ball undergoes simple harmonic motion.

Answer 36

1. -ve sign shows that a is directed to the opp direction to x and always towards eqm position
2. a is directly prop to x since g & r are const