# Phase

Here is an oscillating ball.

Its motion can be described as follows:

- First the ball is at \(v = 0\) at the right
- Then it moves with \(v < 0\) through the centre to the left
- Then it is at \(v = 0\) at the left
- Then it moves with \(v > 0\) through the centre to the right
- Then it repeats...

This information concentrates on what **phase** of the cycle is being executed. It is not concerned with the particulars of amplitude.

Mathematically, the phase is the "\(\omega \; t\)" in:

\(x(t) = A \cos (\omega \; t)\)

The ubiquitous ball undergoes oscillations of amplitude A.

### What is the phase in the following situations?

**What phase is the ball in when:**

\(x = +A\), \(v = 0\)

A) \(0.00 \;\pi \; rad\)

B) \(0.25 \;\pi \; rad\)

C) \(0.50 \;\pi \; rad\)

D) \(1.0 \;\pi \; rad\)

E) \(1.5 \;\pi \; rad\)

F.)\(1.7 \; \pi \;rad\)

**A) \(0.00 \;\pi \; rad\) - Correct!**

B) \(0.25 \;\pi \; rad\) - No. Find a point where \(v = 0\) and \(x = A\) and measure how far it is from the origin

C) \(0.50 \;\pi \; rad\) - No. Find a point where \(v = 0\) and \(x = A\) and measure how far it is from the origin

D) \(1.0 \;\pi \; rad\) - No. Find a point where \(v = 0\) and \(x = A\) and measure how far it is from the origin

E) \(1.5 \;\pi \; rad\) - No. Find a point where \(v = 0\) and \(x = A\) and measure how far it is from the origin

F.)\(1.7 \; \pi \;rad\) - No. Find a point where \(v = 0\) and \(x = A\) and measure how far it is from the origin

**What phase is the ball in when:**

\(x = 0\), \(v < 0\)

A) \(0.00 \;\pi \; rad\)

B) \(0.25 \;\pi \; rad\)

C) \(0.50 \;\pi \; rad\)

D) \(1.0 \;\pi \; rad\)

E) \(1.5 \;\pi \; rad\)

F.)\(1.7 \; \pi \;rad\)

A) \(0.00 \;\pi \; rad\) - No. You need a point where x is zero and v is negative

**B) \(0.25 \;\pi \; rad\) - Correct!**

C) \(0.50 \;\pi \; rad\) - No. You need a point where x is zero and v is negative

D) \(1.0 \;\pi \; rad\) - No. You need a point where x is zero and v is negative

E) \(1.5 \;\pi \; rad\) - No. You need a point where x is zero and v is negative

F.) \(1.7 \; \pi \;rad\) - No. You need a point where x is zero and v is negative