The potential energy of a harmonic oscillator of mass 2 kg in its mean positioin is 5J. If its total energy is 9J and its amplitude is 0.01m, its period will be

Options

A. `(pi/100)s`
B. `(pi)/(50)s`
C. `(pi)/(20)s`
D. `(pi)/10)`s

Answer

To solve the problem step by step, we will use the information given about the harmonic oscillator and apply relevant formulas.

### Step 1: Identify the given values
- Mass (m) = 2 kg
- Potential Energy at mean position (PE) = 5 J
- Total Energy (E) = 9 J
- Amplitude (A) = 0.01 m

### Step 2: Calculate the maximum kinetic energy (KE_max)
The total energy (E) of a harmonic oscillator is the sum of its potential energy (PE) and maximum kinetic energy (KE_max). At the mean position, the potential energy is at its minimum (which is zero), and the kinetic energy is at its maximum.

Using the formula:
\[ E = PE + KE_{max} \]

Substituting the known values:
\[ 9 J = 5 J + KE_{max} \]

Now, solve for KE_max:
\[ KE_{max} = 9 J - 5 J = 4 J \]

### Step 3: Relate kinetic energy to velocity
The maximum kinetic energy can also be expressed in terms of mass and maximum velocity (V_max):
\[ KE_{max} = \frac{1}{2} m V_{max}^2 \]

Substituting the known values:
\[ 4 J = \frac{1}{2} \times 2 kg \times V_{max}^2 \]

This simplifies to:
\[ 4 J = 1 kg \times V_{max}^2 \]
\[ V_{max}^2 = 4 \]
\[ V_{max} = \sqrt{4} = 2 m/s \]

### Step 4: Relate maximum velocity to angular frequency
The maximum velocity in simple harmonic motion is given by:
\[ V_{max} = A \omega \]
Where:
- A = amplitude
- ω = angular frequency

Substituting the known values:
\[ 2 m/s = 0.01 m \times \omega \]

Now, solve for ω:
\[ \omega = \frac{2 m/s}{0.01 m} = 200 rad/s \]

### Step 5: Calculate the period (T)
The period (T) of a harmonic oscillator is related to angular frequency (ω) by the formula:
\[ T = \frac{2\pi}{\omega} \]

Substituting the value of ω:
\[ T = \frac{2\pi}{200 rad/s} \]

Calculating the period:
\[ T = \frac{2\pi}{200} = \frac{\pi}{100} \, seconds \]

### Final Answer
The period of the harmonic oscillator is:
\[ T = \frac{\pi}{100} \, seconds \]

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