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Simple Harmonic Motion. Spring motion. Let’s assume that a mass is attached to a spring, pulled back, and allowed to move on a frictionless surface…. k. m. k. m. k. m. Simple Harmonic Motion (SHM).

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Presentation Transcript
spring motion
Spring motion
  • Let’s assume that a mass is attached to a spring, pulled back, and allowed to move on a frictionless surface…
simple harmonic motion shm

k

m

k

m

k

m

Simple Harmonic Motion (SHM)
  • We know that if we stretch a spring with a mass on the end and let it go, the mass will oscillate back and forth (if there is no friction).
  • This oscillation is called Simple Harmonic Motion, and is actually very easy to understand...
shm dynamics

F = -kx

a

k

m

x

SHM Dynamics
  • At any given instant we know thatF = mamust be true.But in this case F = -kx and F = ma
  • So: -kx = ma
  • F = - kx (Hooke’s Law)
  • Period is proportional to the square root of mass over spring constant
shm dynamics1
SHM Dynamics...

y = Rcos =Rcos(t)

  • But wait a minute...what does angularvelocityhave to do with moving back & forth in a straight line ??

y

1

1

1

2

2

3

3

0

x

4

6

-1

4

6

5

5

shm and velocity and acceleration
SHM and Velocity and Acceleration
  • If you were to plot the position of the mass over time, it would look like a sine wave.
  • The velocity and acceleration would look a little out of phase
problem vertical spring
Problem: Vertical Spring
  • A mass m = 102 g is hung from a vertical spring. The equilibrium position is at y = 0. The mass is then pulled down a distance d = 10 cm from equilibrium and released at t = 0. The measured period of oscillation is T = 0.8 s.
    • What is the spring constant k?

k

y

0

-d

m

t = 0

the simple pendulum

z

L

m

d

mg

The Simple Pendulum...
  • Period is proportional to the square root of the length over gravity
  • This works when the angle is less than 15o
  • The period does not depend on the mass of the object
simple harmonic motion1
Simple Harmonic Motion
  • You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T1.
  • Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T2.

Which of the following is true:

(a)T1 = T2

(b)T1 > T2

(c) T1 < T2

solution
Solution

Standing up raises the CM of the swing, making it shorter!

L2

L1

T1

T2

Since L1 > L2 we see that T1 > T2 .

energy in shm

U

K

E

U

s

-A

0

A

Energy in SHM
  • For both the spring and the pendulum, we can derive the SHM solution by using energy conservation.
  • The total energy (K + U) of a system undergoing SHM will always be constant!
  • This is not surprising since there are only conservative forces present, hence K+U energy is

conserved.