The motion of a 100 g mass tied to a spring is described by the equation `x=25cos(3t+(pi)/(4))cm`. Find (i) the angular velocity `omega`(ii) frquency v (iii) the time period T (iv) the force constant k (v) the amplitude A and (v) the phase angle

Answer

To solve the problem step by step, we will analyze the given equation of motion and extract the required parameters.

### Given:
The motion of a mass tied to a spring is described by the equation:
\[ x = 25 \cos\left(3t + \frac{\pi}{4}\right) \text{ cm} \]

### Step 1: Find the Angular Velocity \(\omega\)
The standard form of the equation for simple harmonic motion (SHM) is:
\[ x = A \cos(\omega t + \phi) \]
Comparing this with the given equation:
- Amplitude \(A = 25 \text{ cm}\)
- Angular frequency \(\omega = 3 \text{ rad/s}\)

Thus,
\[ \omega = 3 \text{ rad/s} \]

### Step 2: Find the Frequency \(f\)
The relationship between angular frequency \(\omega\) and frequency \(f\) is given by:
\[ f = \frac{\omega}{2\pi} \]
Substituting the value of \(\omega\):
\[ f = \frac{3}{2\pi} \approx 0.477 \text{ Hz} \]

### Step 3: Find the Time Period \(T\)
The time period \(T\) is the reciprocal of frequency:
\[ T = \frac{1}{f} = \frac{1}{0.477} \approx 2.095 \text{ seconds} \]

### Step 4: Find the Force Constant \(k\)
The force constant \(k\) of the spring is given by the formula:
\[ k = \omega^2 m \]
Where \(m\) is the mass in kilograms. The mass is given as \(100 \text{ g}\), which is:
\[ m = \frac{100}{1000} = 0.1 \text{ kg} \]
Now substituting the values:
\[ k = (3)^2 \times 0.1 = 9 \times 0.1 = 0.9 \text{ N/m} \]

### Step 5: Find the Amplitude \(A\)
From the equation, we can see that the amplitude \(A\) is:
\[ A = 25 \text{ cm} = 0.25 \text{ m} \]

### Step 6: Find the Phase Angle \(\phi\)
From the equation, the phase angle \(\phi\) is:
\[ \phi = \frac{\pi}{4} \text{ radians} \quad \text{or} \quad 45^\circ \]

### Summary of Results:
(i) Angular velocity \(\omega = 3 \text{ rad/s}\)
(ii) Frequency \(f \approx 0.477 \text{ Hz}\)
(iii) Time period \(T \approx 2.095 \text{ s}\)
(iv) Force constant \(k = 0.9 \text{ N/m}\)
(v) Amplitude \(A = 25 \text{ cm} = 0.25 \text{ m}\)
(vi) Phase angle \(\phi = \frac{\pi}{4} \text{ radians} \text{ or } 45^\circ\)