Transcript simple harmonic motion.

Simple Harmonic
Motion
Periodic Motion
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Acrobat on a trapeze
Child on a swing
Pendulum of a clock
Mass attached to a spring
The direction of the force acting on the mass is always opposite the
direction of the mass’s displacement from equilibrium (x=0).
When the spring is stretched
to the right, the spring force
pulls the mass to the left.
When the spring is
unstretched, the spring force
is zero.
When the spring is
compressed to the left, the
spring force is directed to the
right.
 At the equilibrium point, the velocity reaches
a maximum…
(think about projectile motion…)
 The force that is pulling the spring back
towards the equilibrium point is called the
restoring force.
 Any periodic motion that is the result of a
restoring force is called:
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simple harmonic motion.
 In 1678 Robert Hooke found that most mass-
spring systems obey a simple relationship
between force and displacement. For small
displacements from the equilibrium:
 Felastic =-kx
 Felastic =-kx
 The negative sign indicates that the
direction of the spring force is always
opposite the direction of the mass’s
displacement from equilibrium
 In other words… the negative sign shows
that the spring force will tend to move the
object back to its equilibrium position.
 Felastic =-kx
 Spring force = -(spring constant x displacement)
 SI units of ‘k’ are N/m
 ‘k’ is the spring constant. The value of ‘k’ is different
for each individual spring (much like μ is unique for
two different surfaces).
 A greater value of ‘k’ means a stiffer spring… b/c a
greater force is needed to stretch or compress the
spring.
 A mass of .55 kg attached to a vertical spring is
stretched with an additional force of 2N. The spring
2.0 cm from its original equilibrium position. What is
the spring constant?
 A slingshot consists of a light leather cup attached to
two rubber bands If it takes a force of 32N to stretch
the bands 1.2 cm, what is the equivalent spring
constant of the rubber bands?
 Stretched and compressed springs contain
elastic potential energy.
 When a spring is held in a position that is
completely stretched ( or completely
stretched) the kinetic energy is converted to
elastic potential energy
 Can you make the connection to elastic
potential energy and potential energy?
The Simple Pendulum
 The simple Pendulum will have an increase in
gravitational potential energy as the
displacement increases.
 The restoring force on a pendulum bob is a
component of the bob’s weight
 it is the component of the force that is
perpendicular to the string
 The maximum displacement from equilibrium
position is called the amplitude.
 Amplitude is typically measured as the
degree or the angle between the pendulum’s
equilibrium position and its max
displacement.
 The period is the time it takes for one
complete cycle of motion for the pendulum.
Notice that after the time period ‘T’ the
object will be back where it started.
 The frequency is the inverse of the period…
Frequency is the number of cycles per unit time.
(The unit of frequency is a Hertz)
 What influences the time it takes for a
pendulum to swing?
 The formula for the period of a simple
pendulum in simple harmonic motion:
 So, the question is:
 If I have a pendulum of length
“L,” how does adding mass to
the end influence the period?
 Calculate the period and frequency of a 3.50
m long pendulum at the following locations:
 The North Pole, where g = 9.832 m/s2
 Chicago, where g= 9.803 m/s2
 Jakarta, Indonesia, where g = 9.782 m/s2
 A trapeze artist swings in simple harmonic
motion with a period of 3.8 s. Calculate the
length of the cables supporting the trapeze.
 So – now we can calculate the period of a
mass-spring system…
 T = 2π√m/k
 A mass of .30 kg is attached to a spring and is
set into vibration with a period of .24 s. What
is the spring constant of the spring?
 When a mass of 25 g is attached to a certain
spring, it makes 20 complete vibrations in 4.0s.
What is the spring constant in the spring?
 A spring of spring constant 30.0 N/m is attached to
different masses. And the system is set in motion.
Find the period and frequency of vibration for
masses of the following magnitudes:
 2.3 kg
 15 g
 1.9 kg