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CHAP 9 : SIMPLE HARMONIC MOTION, graph, and mass attached to horizontal…
CHAP 9 : SIMPLE HARMONIC MOTION
Kinematics of SHM
General equation of displacement,x : x = A sin ωt
general equation acceleration, a = – ω² x a= -Aω² sin ωt
general equation velocity, v = ± ω √ A² - x² v = Aω cosωt
EXAMPLES
The up and down motions of the piston
Swinging bob of a pendulum
The oscillation of molecules or atoms in an object
The mass-oscillation in a spring-mass system
Vibrations of atoms in solid
Definition
SHM is the acceleration of a body is directly proportional to its displacement from the equilibrium position but in the opposite direction
Condition: - periodic motion - move back and forth - over the same path
Characteristics of oscillation
Equilibrium position (xθ)
Fixed point (point of equilibrium)
No resultant force acting at this point
Particles at rest lies at this point
Displacement (x)
The distance moved (+ve or –ve sign)
Amplitude (A)
Maximum displacement from the position of equilibrium
Period (T)
The time taken by a particle undergoing one complete oscillation.
Frequency (f)
Number of oscillation performed per unit time.
Relationships
The force is proportional to the displacement of the object.
The force must always act in a direction towards the equilibrium point.
The acceleration is directly proportional to its displacement.
PERIOD
period for simple pendulum,T = 2π√l/g
period for mass attached to spring, From F=-kx and F=ma
a = - kx/m
we know that a = - ω²x
therefore, ω² = k/m and ω=√k/m
for period, T = 2π/ ω so, T= 2π√m/k
THE ENERGY OF SHM
E=K+U
,
K is kinetic energy
K=1/2mv²
v = ± ω √ A² - x²
K = 1/2 mω²(A² - x²)
U is potential energy
U=1/2kx²
From F=-kx and F=ma, for SHM a = - ω²x
ω²=k/m --> k=mω²
Therefore, U = 1/2 mω²x²
U=1/2kx² = 1/2 mω²x² , K=1/2mv² = 1/2 mω² (A² - x²)
E=K+U = 1/2mv² + 1/2kx² = 1/2 mω √ A² - x² + 1/2 mω²x²
As the object starts to move, the elastic potential energy is converted into kinetic energy, becoming entirely kinetic energy at the equilibrium position
the motion starts with all of the energy stored in the spring known as elastic potential energy
GRAPHS OF SHM
graph v
v against t
v against x
graph a
a against t
a against x
graph energy
energy against t
energy against x
graph x against t
SYSTEM OF SHM
Simple Pendulum, loaded spring
ω = √g/l
T = 2π√g/l
T = 2π√m/k
and mass attached to horizontal spring
