A particle having mass 10 g oscilltes according to the equatioi `(x=(2.0cm) sin[100t+pi/3]`. Find a the amplitude the time period and the spring constant b. the position, the velociyt and the acceleration at t=0.

Explanation

To solve the problem step by step, we will break it down into parts as per the question.

### Part (a): Finding the Amplitude, Time Period, and Spring Constant

1. **Identify the given equation:**
The equation of motion is given as:
\[ x = 2 \, \text{cm} \cdot \sin(100t + \frac{\pi}{3}) \]

2. **Determine the Amplitude (A):**
The amplitude is the coefficient of the sine function in the equation. Thus,
\[ A = 2 \, \text{cm} \]

3. **Determine the Angular Frequency (ω):**
From the equation, we can see that:
\[ \omega = 100 \, \text{rad/s} \]

4. **Calculate the Time Period (T):**
The time period \( T \) is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{100} = \frac{\pi}{50} \, \text{s} \]

5. **Calculate the Spring Constant (k):**
The relationship between angular frequency and spring constant is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
Rearranging gives:
\[ k = \omega^2 \cdot m \]
The mass \( m \) is given as 10 g, which needs to be converted to kg:
\[ m = 10 \, \text{g} = 0.01 \, \text{kg} \]
Now substituting the values:
\[ k = (100)^2 \cdot 0.01 = 10000 \cdot 0.01 = 100 \, \text{N/m} \]

### Summary of Part (a):
- Amplitude \( A = 2 \, \text{cm} \)
- Time Period \( T = \frac{\pi}{50} \, \text{s} \)
- Spring Constant \( k = 100 \, \text{N/m} \)

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### Part (b): Finding Position, Velocity, and Acceleration at \( t = 0 \)

1. **Find the Position at \( t = 0 \):**
Substitute \( t = 0 \) into the position equation:
\[ x(0) = 2 \, \text{cm} \cdot \sin(100 \cdot 0 + \frac{\pi}{3}) = 2 \, \text{cm} \cdot \sin(\frac{\pi}{3}) \]
Knowing that \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \):
\[ x(0) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \, \text{cm} \]

2. **Find the Velocity at \( t = 0 \):**
The velocity \( v(t) \) is the derivative of the position function:
\[ v(t) = \frac{dx}{dt} = 2 \cdot 100 \cdot \cos(100t + \frac{\pi}{3}) \]
At \( t = 0 \):
\[ v(0) = 200 \cdot \cos(\frac{\pi}{3}) = 200 \cdot \frac{1}{2} = 100 \, \text{cm/s} \]

3. **Find the Acceleration at \( t = 0 \):**
The acceleration \( a(t) \) is the derivative of the velocity function:
\[ a(t) = \frac{dv}{dt} = -2 \cdot 100^2 \cdot \sin(100t + \frac{\pi}{3}) \]
At \( t = 0 \):
\[ a(0) = -20000 \cdot \sin(\frac{\pi}{3}) = -20000 \cdot \frac{\sqrt{3}}{2} = -10000\sqrt{3} \, \text{cm/s}^2 \]

### Summary of Part (b):
- Position at \( t = 0 \): \( x(0) = \sqrt{3} \, \text{cm} \)
- Velocity at \( t = 0 \): \( v(0) = 100 \, \text{cm/s} \)
- Acceleration at \( t = 0 \): \( a(0) = -10000\sqrt{3} \, \text{cm/s}^2 \)

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