**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

Thus the total mechanical energy (E

**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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