Evans SPH 4U1
Physics Grade 12
Unit 5: Waves
Note 1: Simple Harmonic Motion
Reference: Chapter 4.5
We begin our wave unit with a discussion on Simple Harmonic Motion since waves are an application of Simple Harmonic Motion.
When any motion repeats itself, back and forth, vibrating over the same path, the motion is described as periodic (an example is a spring and a mass). If a mass attached to a spring is pushed towards the spring (compressing it and storing energy in it) then released, the mass attached to the spring oscillates back and forth about the equilibrium position. When the mass stretches or compresses the spring, the spring exerts a force on the mass that acts to "restore" the spring to its equilibrium position. If there is no friction, the system will oscillate forever. Hooke's law describes this restoring force, or F = -kx. Any vibrating system for which the restoring force is directly proportional to the negative value of the displacement x (or one that obeys Hooke's law) is said to be in Simple Harmonic Motion (SHM).
Simple Harmonic Motion Terms::
Displacement The distance x of the object from the equilibrium position
Amplitude The maximum displacement from the equilibrium position.
Period (T) The time it takes for one cycle (or back and forth motion).
Frequency (f) The number of cycles completed per second (SI unit is Hz, or Hertz).
Energy in the Simple Harmonic Oscillator:
Work is done when a spring is stretched or compressed. The elastic potential energy of the spring is given by
Period of a Simple Pendulum:
Period of SHM The period of an object in SHM is dependent upon the stiffness of the spring (related to k) and the mass (m) that is oscillating. The period does not depend upon the amplitude.
Example 1:
An object follows SHM. If the acceleration at any time is given by a = 4x where x is the position, what is the period? (Answer: 3.14)
Example 2:
A mass-spring system undergoing SHM has a maximum energy of 6.00 J. The mass is 0.15kg and the force constant is 250 N/m.
a) What is the amplitude of the vibration? (Answer: 0.22 m)
b) Determine the maximum speed of the mass. (Answer: 8.94 m/s)
c) Calculate the speed of the mass when it is 10.0 cm from the equilibrium position. (Answer: 7.96 m/s)
Example 3:
Two springs with the same spring constant k = 39.5 N/m are connected to two different masses 1 kg and 4 kg. Both springs are pulled 4 cm below the equilibrium point and released at the same time and allowed to oscillate for 4.8 s. How many times were the masses at the same position at the same instant in time? (Not counting the start). Assume that the equilibrium point is the same for both systems. (Answer: 7)
Text Questions:
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