A mass attached to a spring is free to oscillate, with angular velocity `omega`, in a horizontal plane without friction or damping. It is pulled to a distance `x_(0)` and pushed towards the centre with a velocity `v_(0)` at time `t=0`. Determine the amplitude of the resulting oscillations in terms of the parameters `omega, x_(0)and v_(0)`.

Options

A. `sqrt((v_(0)^(2))/(omega^(2))-x_(0)^(2))`
B. `sqrt(omega^(2)v_(0)^(2)+x_(0)^(2))`
C. `sqrt((x_(0)^(2))/(omega^(2))+v_(0)^(2))`
D. `sqrt((v_(0)^(2))/(omega^(2))+x_(0)^(2))`

Explanation

To determine the amplitude of the resulting oscillations of a mass attached to a spring, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
The mass is attached to a spring and oscillates in a horizontal plane with angular velocity \( \omega \). At time \( t = 0 \), it is at a distance \( x_0 \) from the equilibrium position and is pushed towards the center with a velocity \( v_0 \).

2. **Displacement Equation**:
The displacement \( x(t) \) of the mass at any time \( t \) can be expressed as:
\[ x(t) = A \cos(\omega t + \phi) \]
where \( A \) is the amplitude and \( \phi \) is the phase constant.

3. **Initial Displacement**:
At \( t = 0 \):
\[ x(0) = A \cos(\phi) = x_0 \]
This gives us our first equation:
\[ x_0 = A \cos(\phi) \quad \text{(Equation 1)} \]

4. **Velocity Equation**:
The velocity \( v(t) \) is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \]
At \( t = 0 \), the velocity is:
\[ v(0) = -A \omega \sin(\phi) = -v_0 \]
This gives us our second equation:
\[ v_0 = A \omega \sin(\phi) \quad \text{(Equation 2)} \]

5. **Squaring Both Equations**:
From Equation 1, squaring both sides gives:
\[ x_0^2 = A^2 \cos^2(\phi) \]
From Equation 2, squaring both sides gives:
\[ v_0^2 = A^2 \omega^2 \sin^2(\phi) \]

6. **Adding the Two Equations**:
Adding the squared equations:
\[ x_0^2 + \frac{v_0^2}{\omega^2} = A^2 \cos^2(\phi) + A^2 \sin^2(\phi) \]
Using the Pythagorean identity \( \cos^2(\phi) + \sin^2(\phi) = 1 \):
\[ x_0^2 + \frac{v_0^2}{\omega^2} = A^2 \]

7. **Solving for Amplitude \( A \)**:
Rearranging gives:
\[ A^2 = x_0^2 + \frac{v_0^2}{\omega^2} \]
Therefore, the amplitude \( A \) is:
\[ A = \sqrt{x_0^2 + \frac{v_0^2}{\omega^2}} \]

### Final Result:
The amplitude of the resulting oscillations is:
\[ A = \sqrt{x_0^2 + \frac{v_0^2}{\omega^2}} \]