Oscillations Flashcards
periodic motion:
Motion which repeats itself after regular intervals of time
Oscillatory motion:
Motion in which an object moves to and fro in a repetitive manner about a fixed point in a definite interval of time.
What is SHM?
Type of oscillatory motion, in which:
1. The particle moves in a single dimension
2. The particle oscillates to and fro about a fixed mean position (where Fnet=0)
3. The net force on the particle always gets directed towards the equilibrium position
4. The magnitude of the net force is always proportional to the displacement of the particle from the equilibrium position at that instant.
F(net)=
Fnet=−kx
a=
a=−kx/m
a=−ω²x
x(t)=
x(t)=Asin(ωt+ϕ)
Amplitude:
Maximum displacement on either side of the equilibrium position
Time Period:
Time taken to complete one cycle
T=2π/ω=2π(√m/k)
ω=
ω=√m/k
Frequency:
number of oscillations completed in one second.
nyu(freq)=
ν=ω/2π
=½π√k/m
Phase:
State of a particle with respect to its position and direction of motion at that particular instant
Phase=(ωt+ϕ)
ϕ is called ________ of the particle or phase constant.
initial phase
At any instant t, v(t)=
v(t)=Aωcos(ωt +ϕ)
At any position x, v(x)=
v(x)=±ω√A²−x²
velocity has maximum magnitude at
The equilibrium position.
At any instant t, a(t)=
a(t)=ω²A sin(ωt+ϕ)
At any position x, a(x)=
a(x)=ω²x
Acceleration is always directed towards
equilibrium position
The magnitude of acceleration is minimum at
Equilibrium position and maximum at extremes
K=
K=½mω²(A²-x²)
=½mω²A²cos²(ωt+ϕ)
K(max)=
Kmax=½mω²A²
U(x)=
U(x)=½kx²=½mA²ω²sin²(ωt+ϕ)
T.E.=
T.E.=½kA²=½mA²ω²
To understand whether a motion is S.H.M:
Locate the equilibrium position mathematically by balancing all the forces on it.
Displace the particle from the mean position in the direction of oscillation.
Determine net force on it and check if it is towards the mean position.
Express net force as a proportional function of its displacement
If step (c) and step (d) are proved then it is a simple harmonic motion.
Systems executing SHM:
vertical & horizontal spring: T=2π√m/k
Simple Pendulum: T=2π√ℓ/g
v(t)=
-₩Asin(₩t + ø)
a(t)=
-₩²x(t)
