1. Simple Harmonic Motion1

2. Simple Harmonic Motion
Vibrations
Vocal cords when singing/speaking
String/rubber band
Simple Harmonic Motion
Restoring force proportional to displacement
Springs F = -kx 11

3. Question II
A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the magnitude of the acceleration of the block biggest?
1. When x = +A or -A (i.e. maximum displacement)
2. When x = 0 (i.e. zero displacement)
3. The acceleration of the mass is constant +A t -A x 17

4. Potential Energy in Spring
Force of spring is Conservative
F = -k x
W = -1/2 k x2
Work done only depends on initial and final position
Define Potential Energy PEspring = ½ k x2
Force
work x 20

5. ***Energy in SHM***
A mass is attached to a spring and set to motion. The maximum displacement is x=A
Energy = PE + KE = constant!
= ½ k x2 + ½ m v2
At maximum displacement x=A, v = 0
Energy = ½ k A2 + 0
At zero displacement x = 0
Energy = 0 + ½ mvm2
Since Total Energy is same
½ k A2 = ½ m vm2
vm = sqrt(k/m) A x
PES m x
x=0 25

6. Question 3
A mass on a spring oscillates back & forth with simple harmonic motion of amplitude A. A plot of displacement (x) versus time (t) is shown below. At what points during its oscillation is the speed of the block biggest?
1. When x = +A or -A (i.e. maximum displacement)
2. When x = 0 (i.e. zero displacement)
3. The speed of the mass is constant +A t -A x 29

7. Question 4
A spring oscillates back and forth on a frictionless horizontal surface. A camera takes pictures of the position every 1/10th of a second. Which plot best shows the positions of the mass. 1 2 3
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint
EndPoint
Equilibrium
EndPoint 38

8. Springs and Simple Harmonic Motion
X=0
X=A
X=-A
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax 32

9. What does moving in a circle have to do with moving back & forth in a straight line ??
x = R cos q
= R cos (wt)
since q = w t y x -R R  1 2 3 4 5 6 x 8 8 q R 7 7 34

10. SHM and Circles

11. Simple Harmonic Motion:
x(t) = [A]cos(t)
v(t) = -[A]sin(t)
a(t) = -[A2]cos(t)
x(t) = [A]sin(t)
v(t) = [A]cos(t)
a(t) = -[A2]sin(t) OR
xmax = A
vmax = A
amax = A2
Period = T (seconds per cycle)
Frequency = f = 1/T (cycles per second)
Angular frequency =  = 2f = 2/T
For spring: 2 = k/m 36

12. Example
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
Which equation describes the position as a function of time x(t) =
A) 5 sin(wt) B) 5 cos(wt) C) 24 sin(wt)
D) 24 cos(wt) E) -24 cos(wt)
Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 39

13. Example
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
What is the total energy of the block spring system?
Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 43

14. Example
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
What is the maximum speed of the block?
Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 46

15. Example
A 3 kg mass is attached to a spring (k=24 N/m). It is stretched 5 cm. At time t=0 it is released and oscillates.
How long does it take for the block to return to x=+5cm?
Example: Given m,k and A and x(0), choose x(t), v(t), a(t), Calculate v_max and a_max, period, energy, K, U 49

16. Pendulum Motion For small angles T = mg
Tx = -mg (x/L) Note: F proportional to x!
S Fx = m ax
-mg (x/L) = m ax
ax = -(g/L) x
Recall for SHO a = -w2 x
w = sqrt(g/L)
T = 2 p sqrt(L/g)
Period does not depend on A, or m! L T m x mg 37

17. Example: Clock
If we want to make a grandfather clock so that the pendulum makes one complete cycle each sec, how long should the pendulum be?

18. Question 1
Suppose a grandfather clock (a simple pendulum) runs slow. In order to make it run on time you should:
1. Make the pendulum shorter
2. Make the pendulum longer 38

19. Summary Simple Harmonic Motion Springs Pendulum (Small oscillations)
Occurs when have linear restoring force F= -kx
x(t) = [A] cos(wt)
v(t) = -[Aw] sin(wt)
a(t) = -[Aw2] cos(wt)
Springs
F = -kx
U = ½ k x2
w = sqrt(k/m)
Pendulum (Small oscillations)
w = sqrt(L/g) 50