Sure, let's break down the given information and solve each part of the question step by step.
### Given Information:
The displacement [tex]\( y \)[/tex] of a mass vibrating with simple harmonic motion is given by the equation:
[tex]\[ y = 20 \sin(20 \pi t) \][/tex]
Where [tex]\( y \)[/tex] is in millimeters and [tex]\( t \)[/tex] is in seconds.
### i) Amplitude
The amplitude of a simple harmonic motion can be identified as the coefficient of the sine (or cosine) function in the displacement equation. Here, the equation is [tex]\( y = 20 \sin(20 \pi t) \)[/tex].
Therefore, the amplitude [tex]\( A \)[/tex] is:
[tex]\[ A = 20 \text{ millimeters} \][/tex]
### ii) The Period
The period [tex]\( T \)[/tex] of a simple harmonic motion is the time it takes to complete one full cycle. The period is related to the angular frequency [tex]\( \omega \)[/tex] by the formula:
[tex]\[ T = \frac{2 \pi}{\omega} \][/tex]
In the given equation [tex]\( y = 20 \sin(20 \pi t) \)[/tex], the angular frequency [tex]\( \omega \)[/tex] is the coefficient of [tex]\( t \)[/tex] inside the sine function, which is [tex]\( 20 \pi \)[/tex].
Thus:
[tex]\[ \omega = 20 \pi \][/tex]
Now, substitute [tex]\( \omega \)[/tex] into the period formula:
[tex]\[ T = \frac{2 \pi}{20 \pi} = \frac{2 \pi}{20 \pi} = \frac{1}{10} = 0.1 \text{ seconds} \][/tex]
### iii) The Velocity at [tex]\( t = 0 \)[/tex]
The velocity [tex]\( v(t) \)[/tex] in simple harmonic motion is the time derivative of the displacement [tex]\( y(t) \)[/tex].
Given [tex]\( y = 20 \sin(20 \pi t) \)[/tex], let's differentiate [tex]\( y \)[/tex] with respect to [tex]\( t \)[/tex] to find [tex]\( v(t) \)[/tex]:
[tex]\[ y = 20 \sin(20 \pi t) \][/tex]
[tex]\[ v(t) = \frac{dy}{dt} \][/tex]
Using the chain rule:
[tex]\[ v(t) = 20 \cdot 20 \pi \cos(20 \pi t) \][/tex]
[tex]\[ v(t) = 400 \pi \cos(20 \pi t) \][/tex]
To find the velocity at [tex]\( t = 0 \)[/tex], substitute [tex]\( t = 0 \)[/tex] into the velocity equation:
[tex]\[ v(0) = 400 \pi \cos(20 \pi \cdot 0) \][/tex]
[tex]\[ v(0) = 400 \pi \cos(0) \][/tex]
[tex]\[ \cos(0) = 1 \][/tex]
[tex]\[ v(0) = 400 \pi \times 1 = 400 \pi \][/tex]
The velocity at [tex]\( t = 0 \)[/tex] is:
[tex]\[ v(0) = 400 \pi \text{ millimeters per second} \][/tex]
Converting [tex]\( 400 \pi \)[/tex] to a numerical value:
[tex]\[ v(0) \approx 1256.64 \text{ millimeters per second} \][/tex]
### Summary
1. Amplitude: 20 millimeters
2. Period: 0.1 seconds
3. Velocity at [tex]\( t = 0 \)[/tex]: approximately 1256.64 millimeters per second
