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10. Simple Harmonic Motion R FELIX ROOM 216 Simple Harmonic Motion Simple harmonic motion (SHM) is an example of periodic motion in which the restoring force is linear. It is a simple approximation that can be applied to many real world scenarios and allows for a straight forward analysis of these situations. It can be applied to buildings, bridges, and many other objects in the real world.
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10. Simple Harmonic Motion
R FELIX ROOM 216
Simple Harmonic Motion
Simple harmonic motion (SHM) is an example ofperiodic motion in which the restoring force is linear. It isa simple approximation that can be applied to many realworld scenarios and allows for a straight forwardanalysis of these situations. It can be applied tobuildings, bridges, and many other objects in the realworld.
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When a spring is stretched or compressed, it will exert a force in the direction that will restore it to its original shape. This force is called the restoring force.
The restoring force always acts in the oppositedirection that the spring is stretched or compressed. In other words, it acts in the oppositedirection of the applied force.
Fr = Fa
SpringsProfessor Walter Lewin (09:05)
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Important Diagram and Vocabulary
The maximum displacement from the equilibrium position during a cycle is the amplitude, A.
The time it takes for an object to pass through one cycle is the period, T.
Period is related to frequency, f, which is thenumber of cycles per second.
One repetition of periodic motion is called a cycle.
x = +A
x = 0
x = A
Equilibrium PositionCompressed Stretched
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Springs are said to follow Hooke's Law.
where F is the restoring force exerted by the spring (N) x represents the amount of compression or stretch (m) k represents the spring constant (N/m)
The applied force is directly proportional to the amount of stretch or compression of
a spring.
The spring constant, k, is a value that describes the amount of force that is necessary to stretch or compress a spring a given amount. It is basicallya measure of how "stiff" the spring is. The spring constant varies from spring to spring.
Hooke's Law
The negative sign simply accounts for the fact that the displacement and force are in opposite directions we can ignore it.
(16351703)
F = kx
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Note
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Sample Problem
A slingshot has an elastic cord tied to a Yshaped frame. The cord has a spring constant of 1.10 x 103 N/m. A force of 455 N is applied to the cord.
a) How far does the cord stretch?
b) What is the restoring force from the elastic cord?
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Sample Problem
The spring in a typical Hooke's law apparatus has a spring constant of 1.50 N/m and a maximum stretchof 10.0 cm. What is the largest mass that can be placed on the spring without damaging it?
F = kx
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ExampleA piece of rubber is 45 cm long when aweight of 8.0 N hangs from it and is 58 cmlong when a weight of 12.5 N hangs fromit. What is the spring constant of this pieceof rubber?
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Simple Harmonic Motion
An oscillation is a motion that regularly repeats itselfover the same path. Another term for oscillation is"periodic motion."
Objects undergo periodic motion when they experience a restoring force. This is a force that restores an objectto its equilibrium position. The equilibrium position is the position where the object would be if it weren't moving.
A restoring force doesn't need to bring an object to rest in its equilibrium position; it just needs tomake that object pass through the equilibrium position.
Note
Simple harmonic motion can be defined as motion where acceleration is directly proportional to the displacement from its equilibrium position.
Simple harmonic motion is the study of oscillations.
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Simple Harmonic Motion For a Horizontal Spring
• The negative sign indicates that the force is to the left, so it will be pulled back to equilibrium.
• As the block moves back to equilibrium, it is accelerating.
• Even though the force at equilibrium is again zero, it continues to move due to its momentum.
• Notice that the force is now to the right, so it will again be pulled back to equilibrium.
• This motion will go on indefinitely if there is no loss of energy
Restoring force acts toward equilibrium, resulting inoscillation.
This type of oscillation is referred to as simple harmonic motion.
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Simple Harmonic Motion• Oscillation considered SHM if the restoring force is linear• Position oscillates following sinusoidal pattern
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P QO
a
ax
ay
r
A
B
ax = 0
ax = 0
axay
a
Simple Harmonic Motion
The name refers to a type of movement a regular, back and forth motion.
SHM is related to circular motion at constant speed.In this motion, an object moves back and forth on astraight line in such a way that its acceleration is proportional to its displacement from its equilibrium position and is directed toward that position.
Examples of SHM are the vibrations of an object attached to a spring and the vibrations of the bob of a pendulum
As A moves around the circle at constant speed,B moves from P to Q and back again in simple harmonic motion.
The motion of B is the horizontal component of the motion of A, and the acceleration of B at any moment is therefore the horizontal component of the acceleration of A at that moment.
Reminder
Objects undergoing SHM must experience a restoring force. This is a force that restores an object to its equilibrium position.
Simple Harmonic Motion Background Material
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Simple Harmonic Motion FOR A Vertical Spring
• Net force depends only on displacement from the centre of oscillation (in this case the new equilibrium after placing the on the spring)• Independent of gravity• Can be treated same as horizontal SHM since net restoring force is linear
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Graph of SHM
Simple Harmonic Motion
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A mass attached to the end of a spring will oscillatein simple harmonic motion.
The period of the oscillation can be found using thefollowing equation.
m is the mass of the object on the spring (kg) k is the spring constant (N/m)
where
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Example 2
A spring has a spring constant of 43.2 N/m. If a 135 g mass is placed on it and made to vibrate, what will the frequency of the vibration be?
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Sample Problem
A block with a mass of 10.0 kg is placed on the end of a spring that is hung from the ceiling. When the block is attached to the spring, the spring stretches 20.0 cm from its rest position.
a) Determine the spring constant.b) What is the block's period of oscillation?
5.0 cm
equilibrium position
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Sample Problems
1. A block of mass 1.5 kg is attached to the end of a vertical spring of force constant k = 3.00 x 102 N/m. The block is set into motion.
a) What is the frequency of the block's motion? b) What is the maximum amount of stretch? 2. When a family of four people with a mass of 2.00 x 10 2 kg step into their car, the car's springs compress 3.00 cm.
a) What is the spring constant of the car's springs? (Assume they act like a single spring.) b) What are the period and frequency of the car after hitting a bump?
3. A small insect of mass 0.30 g is caught in a spider web. The web vibrates with a frequency of 15 Hz.
a) What is the value of the spring constant for the web? b) At what frequency would you expect the web to vibrate if an insect of mass 0.10 g were trap ped?
Simple Harmonic Motion
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A pendulum in a grandfather clock undergoes SHM: it travels back and forth, back and forth, back and forth....
In the case of a grandfather clock, the pendulum's equilibrium position is when it is hanging straight down.
When it is swinging, gravity exerts a restoring force: as the pendulum swings, the force of gravity pulls on the pendulum, so that it eventuallyswings back down and passes through its equilibrium position.
It only remains in its equlibrium position for an instant, and then swings back the other way.
Simple Pendulums
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History Link
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From the green triangle we get:
The restoring force is equal to the component of the weight of the bob
that is tangent to the arc.
NoteThe expression
is equal to the spring constant, k, in Hooke's law.
Note
(McGraw Hill Text, page 610)Fgsin θ
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Replace the spring constant, k, in the equation for the period of a mass and spring system and then simplifythe expression.
The equation is only valid if the angle between the equilibrium line and the pendulum string isless than 15o.
Note
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Sample Problem
1. a) Find the period of a pendulum with a 2.45 kg bob and having a length of 1.36 m. b) By what amount would you have to increase the length in order to double the period?
2. Estimate the length of the pendulum in a grand father clock that ticks once a second.
page 608, PP 1 and 2
page 614, PP 5, 6, 8
Textbook
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The Energy of Simple Harmonic Motion
The graph of Hooke's law can be used to determine the quantity of potential energy stored in a spring. The area under the graph is equal to the amount of potential energy stored in a spring.
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Elastic Potential Energy The elastic potential energy of a perfectly elasticmaterial is one half the product of the spring constant and the square of the length of extension or compression.
Quantity Units
Note
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Hooke's law and the concept of elastic potential energy can apply to elastic objects other thansprings. Some elastic devices include:
• bows and arrows• sling shots• bungee cords
Example
An archer does work on the bow by pulling the string and bending the bow. The bow stores elas tic energy that can then be transformed into the kinetic energy of an arrow.
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Elastic Potential Energy andKinetic Energy
Let's consider the simple situation of a mass attachedto a spring of constant k, lying on a frictionless surface.
maximum extension
KE = 0 J(the block is not moving)
equilibrium position
Ee = 0(x = 0 m)
Maximumvelocity
KE = 0 J(the block is not moving)
maximum compression
NoteAt any instant, the total energy of
the system will be given by
ET = Ee + KE
T
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At the maximum displacement, the total mehanical energy is given by:
At the equilibrium position, the total mehanical energy is given by:
Therefore,
and
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Simple Harmonic Motion @ www.ThePhysicsCafe.com
(6:27)
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Sample Problem1. A 0.30 kg mass vibrates at the end of a hori zontal spring along a frictionless surface. If the spring constant is 45 N/m, and the maxi mum displacement of the mass is 0.080 m, what is the maximum speed of the mass?
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Sample Problem2. An object vibrates at the end of a vertical spring (121 N/m). If the potential energy of the object is 2.2 J, what is its maximum displacement?
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Sample Problem3. A 0.40 kg mass vibrates are the end of a horizontal spring along a frictionless surface reaching a maxi mum speed of 0.50 m/s. If the maximum displace ment of the mass is 0.11 m, what is the spring con stant?
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Sample Problem4. An object vibrates at the end of a vertical spring that has a spring constant of 25 N/m. If the maxi mum speed of the object is 0.15 m/s, and its maxi mum displacement is 0.11 m, what is the mass of the object?
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Horizontal Elastic Collisions
1.
2.
page 261, PP 38 40
page 296, PP 9 11
Textbook
Attachments
Hooke's Law, Simple Harmonic Oscillator
Simple Harmonic Motion @ www.ThePhysicsCafe.com
IBPH Episode #7 Simple Harmonic Motion ﴾Part 1﴿
Simple Harmonic Motion
Simple Harmonic Motion Background Material
