1. Chapter 11: Vibrations and Waves Section 1: Simple Harmonic Motion
St. Augustine Preparatory School

2. Periodic Motion
Periodic motion is any type of motion repeated in equal intervals of time

3. Periodic Motion of a Mass on a Spring

4. Online Tutorial

5. At the equilibrium position, the object will be at a maximum speed
At the equilibrium position, the object will be at a maximum speed. At this point, the acceleration of the object is 0 m/s2.

6. At the maximum displacements, the spring force and acceleration are at a maximum. This means that both are the highest that they can possibly be for the system.

7. Without friction, this system will oscillate indefinitely (forever).
Oscillate: move or swing back and forth at a regular speed.

8. Damping Force Given real world systems, friction is unavoidable.
In the system of the mass and the spring, friction is known as the damping force.

9. Hooke’s Law (1678)
Felastic = -kx Felastic: spring force, sometimes called restoring force (unit: N) k: spring constant (unit: N/m) x: displacement (unit: m) Each spring has its own spring constant, which is a measure of the stiffness of a spring.

10. Simple Harmonic Motion
Simple Harmonic Motion describes any periodic motion that is the result of a restoring force that is proportional to the displacement.
A back-and-forth motion that is repeated over the same path

11. Example
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0cm from its original equilibrium position, what is the spring constant?

12. Solution Fspring = mg = (0.55kg)(9.81m/s2) = 5.3955 N k = ?
If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0cm from its original equilibrium position, what is the spring constant?
Fspring = mg = (0.55kg)(9.81m/s2) = N
k = ?
x = m
k = Fspring / x = ( N)/(-0.020m)
k = N/m = 270 N/m
(spring constants are always positive, so our answer looks reasonable)

13. Practice Problems
A load of 45 N is attached to a spring that is hanging vertically and stretches the spring 0.14m. Determine the spring constant.
A mass of 15 kg is hung vertically from a spring that has a spring constant of 105 N/m. How far will the spring stretch.

14. Solutions
A load of 45 N is attached to a spring that is hanging vertically and stretches the spring 0.14m. Determine the spring constant.
Fspring = -kx
-(Fspring / x) = k
-(45N/-0.14m) = k
321.43N/m = k
3.2x102 N/m = k
A mass of 15 kg is hung vertically from a spring that has a spring constant of 105 N/m. How far will the spring stretch.
Fspring / -k = x
(15kg*9.81m/s2) / -105N/m = x
m = x
-1.4 m = x

15. Stretched/Compressed Springs Have Elastic Potential Energy
When springs are stretched or compressed, there is energy within the system. If the spring is compressed and held down, or stretched and held out, all of the energy will be potential energy. Once released, this energy can be converted to kinetic energy.
Recall, conservation of mechanical energy.

17. Pendulums Notes: - The pivot of the pendulum
is considered frictionless so no
energy is lost.
- The mass at the bottom is called
a bob.
The rod or string the bob hangs
from is considered to be massless

18. Formula
The restoring force for a pendulum can be found using the following:
Fg,x = Fgsinθ , where
Fg,x is the restoring force (unit: N)
Fg is the force of gravity (unit: N), recall Fg = mg
sinθ is the angle of the displacement (unit: °)

19. Example
At a certain point in a pendulum, a 1.20 kg bob requires a force of 4.20 N to restore it to its equilibrium position. What angle is the bob currently at?

20. Pendulums as Simple Harmonic Motion
When the angle of displacement of a pendulum is relatively small (<15°), the motion of a pendulum is very similar to simple harmonic motion.
Angles greater than this begin to differ from simple harmonic motion.

21. Both pendulums and spring systems share many things in common.