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Chapter 2
Simple Harmonic Motion (SHM)
Learning Outcomes
Simple Harmonic Motion (amplitude, frequency, displacement, velocity, acceleration)
Conservation of energy in SHM Simple pendulum and spring Damped harmonic motion
Simple Harmonic Motion Position x vs. time t Definition of period T Definition of amplitude A
Simple Harmonic Motion Definitions of Terms
• Amplitude = A = the maximum displacement of the moving object from its equilibrium position.
• (unit = m)
• Period = T = the time it takes the object to complete one full cycle of motion.
• (unit = s)
• Frequency = f = the number of cycles or vibrations per unit of time.
• (unit = cycles/s = 1/s = Hz = hertz)
Simple Harmonic MotionMotion described by this equation:
is called simple harmonic motion (SHM). It is: periodic (repeats itself in time) oscillatory (takes place over a limited spatial
range)
tAx cosdisplacement (m)
amplitude (m)time (s)
angular frequency (rad/s)
SHM: Reference Circle Representation
A vector of magnitude
A rotates about the
origin with an angular
velocity
The x component of
the vector represents
the displacement.
t
A
A cos t
X
Y
SHM: Frequency
Since there are 2 radians in each trip (“cycle”) around the reference circle, the “cycle” frequency is related to the angular frequency by
SI units of “cycle” frequency, f:
cycles / s = Hertz (Hz)
2
or 2 ff
SHM: Velocity
We can calculate the
velocity from the
reference circle
representation: t
A
X
Y
tvT
- vT sin t
tAv
tAtv
Arv
T
T
sin
sinsin
SHM: Acceleration
tAa
taa
Ara
C
C
cos
cos
2
22
t
X
Y
taC
-aC cos t
General EquationsDisplacement from Equilibrium:
x(t) =
Velocity: v(t) = =
Acceleration: a(t) = =
Simple Harmonic Motion
dt
dx
dt
dv
A cos ( t + )
A sin ( t + )
A cos ( t + )
SHM Systems
The Period of a Mass on a SpringSince the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that
Substituting the time dependencies of a and x gives
The Period of a Mass on a Spring
Therefore, the period is
Energy Conservation in Oscillatory Motion
In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring:
Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:
Energy Conservation in Oscillatory Motion
As a function of time,
So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.
The Pendulum
A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.
Energy Conservation in Oscillatory Motion
This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.
The Pendulum
A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).
The angle it makes with the vertical varies with time as a sine or cosine.
The Pendulum
Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).
The Pendulum
However, for small angles, sin θ and θ are approximately equal.
The PendulumSubstituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:
The Pendulum
A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:
The PendulumIn this case, it can be shown that the period depends on the moment of inertia:
Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,
Damped OscillationsIn most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed:
This causes the amplitude to decrease exponentially with time:
Damped OscillationsThis exponential decrease is shown in the figure:
Damped OscillationsThe previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.
Summary of Chapter 2• Period: time required for a motion to go through a complete cycle
• Frequency: number of oscillations per unit time
• Angular frequency:
• Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.
Summary of Chapter 2• The amplitude is the maximum displacement from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
Summary of Chapter 2
• Acceleration as a function of time:
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
Summary of Chapter 2
• Potential energy as a function of time:
• Kinetic energy as a function of time:
• A simple pendulum with small amplitude exhibits simple harmonic motion
Summary of Chapter 2• Period of a simple pendulum:
• Period of a physical pendulum: