Lecture 18:Lecture 18: Elasticity and Oscillations I Elasticity and Oscillations I

Simple Harmonic Motion: Definition

Springs: Forces

Springs: Energy

Simple Harmonic Motion: Equations

Simple Harmonic MotionSimple Harmonic Motion

VibrationsVocal cords when singing/speakingString/rubber band

Simple Harmonic MotionRestoring force proportional to displacementSprings: F = -kx

Springs: ForcesSprings: Forces Hooke’s Law:Hooke’s Law: The force exerted by a spring is

proportional to the distance the spring is stretched or compressed from its relaxed position.

FX = -k x Where x is the displacement from the relaxed position and k is the

constant of proportionality.

relaxed position (equilibrium position)

FX = 0+x

x=0

Springs: ForcesSprings: Forces Negative displacement:

Positive forcePositive acceleration

Zero displacement (at equilibrium):Zero forceZero acceleration

Positive displacement:Negative forceNegative acceleration

+x

Springs: EnergySprings: Energy Force of spring is Conservative:

F = -k x W = ½ k x2

Work done only depends on initial and final position.Potential Energy: Uspring = ½ k x2

A mass oscillating on a spring: Energy = U + K = constant

½ k x2 + ½ m v2 = total energym

xx=0

0 x

U

Maximum Negative displacement:Maximum potential energyZero kinetic energy

Zero displacement (at equilibrium):Zero potential energyMaximum kinetic energy

Maximum Positive displacement:Maximum potential energyZero kinetic energy

+x Springs: EnergySprings: Energy

Simple Harmonic MotionSimple Harmonic Motion

x(t) = [A]cos(t)v(t) = -[A]sin(t)a(t) = -[A2]cos(t)

x(t) = [A]sin(t)v(t) = [A]cos(t)a(t) = -[A2]sin(t)

xmax = A

vmax = A

amax = A2

Period = T (seconds per cycle)Frequency = f = 1/T (cycles per

second)Angular frequency = = 2f = 2/TFor spring: 2 = k/m

OR

Spring ExampleSpring Example A mass of m = 4 kg is attached to a spring with k = 16 N/m and

oscillates with an amplitude A = 0.15 m. What is the magnitude of the acceleration of the mass when the potential energy stored in the spring is equal to the kinetic energy of the mass?

First, use energy to find x where U = K:» U + K = constant

Second, use Hooke’s Law to find force on the mass:» F = -k x

Third, use Newton’s Second Law to find acceleration of mass:» F = m a

Spring ExampleSpring Example A mass of m = 4 kg is attached to a spring with k = 16 N/m and

oscillates with an amplitude A = 0.15 m. What is the magnitude of the acceleration of the mass when the potential energy stored in the spring is equal to the kinetic energy of the mass?

First, use energy to find x where U = K:» U + K = constant» U + K = ½ k A2 (since total energy is ½ k A2)» U + U = ½ k A2 (since K = U when they are equal)» k x2 = ½ k A2 (since U = ½ k x2 when they are equal)» x2 = ½ A2 (simplify)

2

21 Ax = 0.106 m

Spring ExampleSpring Example A mass of m = 4 kg is attached to a spring with k = 16 N/m and

oscillates with an amplitude A = 0.15 m. What is the magnitude of the acceleration of the mass when the potential energy stored in the spring is equal to the kinetic energy of the mass?

Second, use Hooke’s Law to find force on the mass:

» F = k x (we are only concerned with magnitude)

2

21 AkF = 1.70 N

Spring ExampleSpring Example A mass of m = 4 kg is attached to a spring with k = 16 N/m and

oscillates with an amplitude A = 0.15 m. What is the magnitude of the acceleration of the mass when the potential energy stored in the spring is equal to the kinetic energy of the mass?

Third, use Newton’s Second Law to find acceleration of mass:

» F = m a (we are only concerned with magnitude)

2

21 A

mka = 0.424 m/s2

Young’s ModulusYoung’s Modulus Spring F = k x

What happens to “k” if cut spring in half?1) ½ 2) same 3) 2

k is inversely proportional to length! Define

Strain = L / LStress = F/A

NowStress = Y Strain F/A = Y L/Lk = Y A/L from F = k x

Y (Young’s Modules) independent of L

15

What does moving in a circle have to do with moving back & forth in a straight line ??

y

x

-R

R

0

1 12 2

3 3

4 45 5

6 62

R

8

7

8

7

23

x

Moviex = R cos = R cos (t)

since = t

46

(A) (B) (C)

(A) (B) (C)