Simple Harmonic Simple Harmonic Motion & ElasticityMotion & Elasticity

Chapter 10Chapter 10

Elastic Potential EnergyElastic Potential Energy

►What is it?What is it? Energy that is stored in elastic materials as a Energy that is stored in elastic materials as a

result of their stretching.result of their stretching.

►Where is it found?Where is it found? Rubber bandsRubber bands Bungee cordsBungee cords TrampolinesTrampolines SpringsSprings Bow and Arrow Bow and Arrow Guitar stringGuitar string Tennis RacquetTennis Racquet

Hooke’s LawHooke’s Law► A spring can be stretched or compressed A spring can be stretched or compressed

with a force.with a force.► The force by which a spring is The force by which a spring is

compressed or stretched is proportional compressed or stretched is proportional to the magnitude of the displacement (to the magnitude of the displacement (FF x).x).

► Hooke’s Law:Hooke’s Law:FFelasticelastic = -k = -kxx

Where:Where:

k = spring constant = stiffness of k = spring constant = stiffness of spring (N/m)spring (N/m)

x = displacementx = displacement

Hooke’s LawHooke’s Law►What is the graphical What is the graphical

relationship between the relationship between the elastic spring force and elastic spring force and displacement?displacement?

FFelasticelastic = -k = -kxx

Displacement

Slope = k

Hooke’s LawHooke’s Law

► A force acting on a spring, A force acting on a spring, whether stretching or whether stretching or compressing, is always compressing, is always positive.positive. Since the spring would prefer to be Since the spring would prefer to be

in a in a ““relaxedrelaxed”” position, a negative position, a negative “restoring”“restoring” force will exist whenever force will exist whenever it is deformed. it is deformed.

The restoring force will always The restoring force will always attempt to bring the spring and attempt to bring the spring and any object attached to it back to any object attached to it back to the equilibrium position.the equilibrium position.

Hence, the restoring force is always Hence, the restoring force is always negative.negative.

Example 1:Example 1:

► A 0.55 kg mass is attached to a vertical spring. If the A 0.55 kg mass is attached to a vertical spring. If the spring is stretched 2.0 cm from its original position, spring is stretched 2.0 cm from its original position, what is the spring constant?what is the spring constant?

► Known:Known:m = 0.55 kgm = 0.55 kg

x = -2.0 cmx = -2.0 cm

g = 9.81 m/sg = 9.81 m/s22

► Equations:Equations:FFnetnet = 0 = F = 0 = Felasticelastic + F + Fg g (1)(1)

FFelasticelastic = -kx = -kx (2)(2)

FFg g = -mg= -mg (3)(3)

Substituting 2 and 3 into 1 yields:Substituting 2 and 3 into 1 yields:

k = -mk = -mgg/x/x

k = -(0.55 kg)(9.81 m/sk = -(0.55 kg)(9.81 m/s22)/-(0.020 m))/-(0.020 m)

k = 270 N/mk = 270 N/m Fg

Felastic

Elastic Potential Energy in a Elastic Potential Energy in a SpringSpring

►The force exerted to put a spring The force exerted to put a spring in tension or compression can be in tension or compression can be used to do work. Hence the spring used to do work. Hence the spring will have will have Elastic Potential EnergyElastic Potential Energy..

►Analogous to kinetic energy:Analogous to kinetic energy:

PEPEelasticelastic = ½ kx = ½ kx22

Example 2:Example 2:

► A 0.55 kg mass is attached to a vertical A 0.55 kg mass is attached to a vertical spring with a spring constant of 270 N/m. If spring with a spring constant of 270 N/m. If the spring is stretched 4.0 cm from its the spring is stretched 4.0 cm from its original position, what is the Elastic Potential original position, what is the Elastic Potential Energy?Energy?

► Known:Known:m = 0.55 kgm = 0.55 kgx = -4.0 cmx = -4.0 cmk = 270 N/mk = 270 N/mg = 9.81 m/sg = 9.81 m/s22

► Equations:Equations:PEPEelasticelastic = ½ kx = ½ kx22

PEPEelasticelastic = ½ (270 N/m)(0.04 m) = ½ (270 N/m)(0.04 m)22

PEPEelasticelastic = 0.22 J = 0.22 JFg

Felastic

►What is the difference in the elastic What is the difference in the elastic potential energy of the system when the potential energy of the system when the deflection is maximum in either the positive deflection is maximum in either the positive or negative direction?or negative direction?

Elastic Potential EnergyElastic Potential Energy►What is area under the curve?What is area under the curve?

Displacement

A = ½ bhA = ½ xFA = ½ xkxA = ½ kx2

Which you should see equals the elastic potential energy

What is Simple Harmonic What is Simple Harmonic Motion?Motion?

►Simple harmonic motion exists Simple harmonic motion exists whenever there is a restoring force whenever there is a restoring force acting on an object.acting on an object. The restoring force acts to bring the object The restoring force acts to bring the object

back to an equilibrium position where the back to an equilibrium position where the potential energy of the system is at a potential energy of the system is at a minimum.minimum.

Simple Harmonic Motion & Simple Harmonic Motion & SpringsSprings

► Simple Harmonic Motion:Simple Harmonic Motion: An oscillation around an equilibrium position An oscillation around an equilibrium position

will occur when an object is displaced from will occur when an object is displaced from its equilibrium position and released.its equilibrium position and released.

For a spring, the restoring force F = -kx.For a spring, the restoring force F = -kx.► The spring is at equilibrium The spring is at equilibrium

when it is at its relaxed length.when it is at its relaxed length.

(no restoring force)(no restoring force)► Otherwise, when in tension or Otherwise, when in tension or

compression, a restoring compression, a restoring

force will exist.force will exist.

Simple Harmonic Motion & Simple Harmonic Motion & SpringsSprings

► At maximum displacement (At maximum displacement (++ x):x): The Elastic Potential Energy The Elastic Potential Energy

will be at a maximumwill be at a maximum The force will be at a The force will be at a

maximum.maximum. The acceleration will be at a The acceleration will be at a

maximum.maximum.► At equilibrium (x = 0):At equilibrium (x = 0):

The Elastic Potential Energy The Elastic Potential Energy will be zerowill be zero

Velocity will be at a maximum.Velocity will be at a maximum. Kinetic Energy will be at a Kinetic Energy will be at a

maximummaximum The acceleration will be zero, The acceleration will be zero,

as will the unbalanced as will the unbalanced restoring force.restoring force.

10.3 10.3 Energy and Simple Harmonic Energy and Simple Harmonic MotionMotion

A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring?

Example 3 Changing the Mass of a Simple Harmonic Oscilator

10.3 10.3 Energy and Simple Harmonic Energy and Simple Harmonic MotionMotion

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Simple Harmonic Motion of Simple Harmonic Motion of SpringsSprings

► Oscillating systems such as that of a spring Oscillating systems such as that of a spring follow a sinusoidal wave pattern.follow a sinusoidal wave pattern.

► Harmonic Motion of Springs – 1Harmonic Motion of Springs – 1► Harmonic Motion of Springs (Concept Simulator)Harmonic Motion of Springs (Concept Simulator)

Frequency of OscillationFrequency of Oscillation

► For a spring oscillating system, the For a spring oscillating system, the frequency and period of oscillation can be frequency and period of oscillation can be represented by the following equations:represented by the following equations:

► Therefore, if the mass of the spring and the spring Therefore, if the mass of the spring and the spring constant are known, we can find the frequency and constant are known, we can find the frequency and period at which the spring will oscillate.period at which the spring will oscillate.

Large k and small mass equals high frequency Large k and small mass equals high frequency of oscillation of oscillation (A small stiff spring)(A small stiff spring)..

12

2

k mf and T

m k

Harmonic Motion & Simple Harmonic Motion & Simple The PendulumThe Pendulum

► Simple Pendulum: Consists of a massive object Simple Pendulum: Consists of a massive object called a bob suspended by a string.called a bob suspended by a string.

► Like a spring, pendulums go through Like a spring, pendulums go through

simple harmonic motion as follows.simple harmonic motion as follows.

Where:Where:

T = periodT = period

l = length of pendulum stringl = length of pendulum string

g = acceleration of gravityg = acceleration of gravity

► Note: Note: 1.1. This formula is true for only small angles of This formula is true for only small angles of θθ..

2.2. The period of a pendulum is independent of its The period of a pendulum is independent of its mass.mass.

Conservation of ME & The Conservation of ME & The PendulumPendulum

► In a pendulum, Potential Energy is converted into In a pendulum, Potential Energy is converted into Kinetic Energy and vise-versa in a continuous repeating Kinetic Energy and vise-versa in a continuous repeating pattern.pattern.

PE = mghPE = mgh KE = ½ mvKE = ½ mv22

MEMET T = PE + KE= PE + KE

MEMETT = Constant = Constant

► Note: Note: 1.1. Maximum kinetic energy is achieved at the lowest point of Maximum kinetic energy is achieved at the lowest point of

the pendulum swingthe pendulum swing..

2.2. The maximum potential energy is achieved at the top of The maximum potential energy is achieved at the top of the swing.the swing.

3.3. When PE is max, KE = 0, and when KE is max, PE = 0.When PE is max, KE = 0, and when KE is max, PE = 0.

Key IdeasKey Ideas

► Elastic Potential Energy is the energy Elastic Potential Energy is the energy stored in a spring or other elastic material.stored in a spring or other elastic material.

► Hooke’s Law: The displacement of a spring Hooke’s Law: The displacement of a spring from its unstretched position is from its unstretched position is proportional the force applied.proportional the force applied.

► The slope of a force vs. displacement The slope of a force vs. displacement graph is equal to the spring constant.graph is equal to the spring constant.

► The area under a force vs. displacement The area under a force vs. displacement graph is equal to the work done to graph is equal to the work done to compress or stretch a spring.compress or stretch a spring.

Key IdeasKey Ideas

►Springs and pendulums will go through Springs and pendulums will go through oscillatory motion when displaced oscillatory motion when displaced from an equilibrium position.from an equilibrium position.

►The period of oscillation of a simple The period of oscillation of a simple pendulum is independent of its angle pendulum is independent of its angle of displacement of displacement (small angles)(small angles) and and mass.mass.

► Conservation of energy: Energy can be Conservation of energy: Energy can be converted from one form to another, but it is converted from one form to another, but it is always conserved.always conserved.