Oscillations Flashcards
Defn of oscillation
Is a special periodic motion in which the oscillator moves to and fro abt an eqm position
Defn of simple harmonic motion
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
Eqn for simple harmonic motion
a = - ω^2 x
Defn of angular frequency
Rate of change of phase angle of the oscillation and is equal to the product of 2π and its frequency
Eqn of angular freq
ω = dΘ / dt (rad s-1)
= 2π / T
= 2πf
Defn of amplitude
Mag of the max displacement of the particle from its eqm position
For a pendulum, where is the max/0 ACC & speed and where is acc directed at at each pt
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
Eqn for angular freq for spring mass system
ω = √ (k/m) = 2πf
Eqn for angular freq for simple pendulum system
ω = √ (g/L) = 2πf
Eqn for instantaneous disp
when obj starts from eqm position
when obj starts from amp
X = Xo sin(ωt) X = Xo cos(ωt)
graph for 2 instantaneous disp
sin & cos graph with the max n min pts at +Xo & -Xo
Eqn for velocity
from eqm
from amp
Just differentiate
X = ωXo cos(ωt)
X = -ωXo sin(ωt)
Eqn for Vmax
±ωXo
Eqn for acc
from eqm
from amp
a = - ω^2 Xo sin(ωt) a = - ω^2 Xo cos(ωt)
Draw V-Xo graph
check notes- circle shape
Draw a - Xo graph
straight line graph with negative grad
grad = - ω^2
Defn of Damping
damping is the process whereby energy is removed from an oscillating system
defn of damped oscillation
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
Types of damping & defn of each
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
Types of damping on x-t graph
check notes
Application: damping in car suspension system
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
Defn of forced oscillations
oscillations are maintained by an external driving force. The system oscillates at the freq of the driving force
Defn of resonance
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.
What does amp of forced oscillation depend (2)
- damping of the system
increased damping –> amp of resonance decrease –> resonance freq decreases as natural period of oscillating system increases - 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
how does the amp-driver freq graph look like for natural, light, heavy damping
check notes
What happens to the light damping graph when mass is increased
mass increases –> freq decreases –> inertia increases –> comp??? increases
How does the expression (eqn) / graph show simple harmonic motion
check notes
eqn for max vel
v = ωx0
eqn for acc
a = - ω^2 x
Eqn for max Ek/Pe/total energy
1/2 m ω^2 Xo^2
Eqn for vel (in formulae list)
v = ±ω√(X0^2 - X^2)
What happens to F-x (OR a-x) graph when there’s friction
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
At which pt does obj leave the ‘platform’ - how to find?
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
How to calc total spring constant for springs in series & parallel
Parallel: Knet = K1 + K2
Series: 1/Knet = 1/K1 + 1/K2
How to convert a-t to x-t graph
literally flip it abt x-axis
Wrt the exp provided, show that the ball undergoes simple harmonic motion.
- -ve sign shows that a is directed to the opp direction to x and always towards eqm position
- a is directly prop to x since g & r are const
