CH 6 Lecture 2 Conservation of Energy

I. Elastic Potential EnergyA. Pushing/Pulling on a spring

1) Stretching or compressing an object from a preferred state requires work

2) The force opposing is elastic if the object snaps back into shape

3) We give the object Elastic Potential Energy

B. Springs

1) Spring constant = k (Tells us how stiff the spring is)

2) Stiffer the spring, larger k

3) Force of spring is proportional to distance it is stretched/compressed

4) Hooke’s Law F = -kx (x = distance stretched or compressed)

a) -k means spring force is always opposing its motion

b) Push on the spring, it pushes back

c) Pull on the spring, it pulls back

C. Elastic Potential Energy = 2

2

1kx

2)( kxxkxFdPEW ½ comes from avg force being half max. force

22

kg/s k - kms

kgm

km- N kx - F

k = 2 kg/s, x = 5 m, F?, W?, PE?

Average Force

II. Conservation of EnergyA) Total Energy of a system remains the same unless it is acted on by an external

force KE + PE = constant

B) Pendulum

1) Side1: KE = 0, Total E = PE

2) Bottom: PE = 0, KE = Total E

3) Side2: KE = 0, Total E = PE

4) Work to get this started, after that W = 0

a) Input E into the system

b) ET = KE + PE = constant

c) Sides: Initial Work gives us PE

d) Bottom: Gravity moves bob down (KE)

e) F = tension = centripetal force; perpendicular to motion, W = Fd = 0

f) Friction (air resistance) does small work, eventually stopping bob

g) In a vacuum, the pendulum would keep swinging forever

C) Could we describe the pendulum with Newton’s laws?

1) Velocity is continually changing

2) Calculations would be hard

F

D) Use Conservation of Energy to Solve hard Mechanics problems

1) Pendulum, m = 0.5 kg h = 12 cm v at bottom?

At top, ET = PE = mgh = (0.5kg)(9.8m/s)(0.12m) = 0.588 J

At bottom, ET = KE = ½ mv2 = (0.5)(0.5kg)v2 = 0.588 J v2 = 2.35 m2/s2

v = 1.53 m/s

E) Energy on a mountain

KE = ½ mv2 If KE decreases, v decreases

III. Springs and Harmonic MotionA) Simple Harmonic Motion: repetitive motion with constant conversion KE/PE

1) Pendulum

2) Mass at the end of a spring

3) Add E with Initial amount of Work

B) Plotting Position vs. Time

1) “Harmonic Function” (sin or cos)

2) One complete cycle = T, period

3) Frequency = # cycles per second (Hz = s-1)

4) Amplitude = max. distance from starting pt.

5) Frequency for

a) Loose/Tight spring?

b) Large/Small mass?

KE = 0

PE = 0

KE = 0

C) Restoring Forces lead to Harmonic Motion

1) Wants to bring mass back to starting position

2) If F is proportional to d, get harmonic motion (F = -kx)

3) What is restoring force for a simple pendulum?

D) Case of a vertical spring/mass system

1) Force of Gravity pulling down is constant

2) Restoring force pulling up varies

FT = FR + FG

3) Equilibrium point is lower than without gravity

4) Harmonic Motion just like horizontal setup

5) FT is still proportional to x

6) PE = gravity + elastic potential energy

Animations