A body weighing 10g has a velocity of `6cms^(-1)` after one second of its starting from mean position. If the time period is 6 seconds. Find the kinetic energy, potential energy and the total energy.

Answer

To solve the problem step by step, we will find the kinetic energy, potential energy, and total energy of the body in simple harmonic motion.

### Step 1: Identify the given values
- Mass of the body (m) = 10 g = 0.01 kg (since we need to convert grams to kilograms)
- Velocity (v) after 1 second = 6 cm/s = 0.06 m/s (convert cm/s to m/s)
- Time period (T) = 6 s

### Step 2: Calculate angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the time period:
\[ \omega = \frac{2\pi}{6} = \frac{\pi}{3} \text{ rad/s} \]

### Step 3: Calculate the amplitude (A)
The velocity in simple harmonic motion is given by:
\[ v = A \omega \cos(\omega t) \]
At \( t = 1 \) second, we can substitute the known values:
\[ 0.06 = A \left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ 0.06 = A \left(\frac{\pi}{3}\right) \left(\frac{1}{2}\right) \]
\[ 0.06 = \frac{A \pi}{6} \]
Now, solving for \( A \):
\[ A = \frac{0.06 \times 6}{\pi} = \frac{0.36}{\pi} \text{ m} \]

### Step 4: Calculate kinetic energy (KE)
The kinetic energy is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values:
\[ KE = \frac{1}{2} \times 0.01 \times (0.06)^2 \]
Calculating:
\[ KE = \frac{1}{2} \times 0.01 \times 0.0036 = 0.000018 \text{ J} = 18 \text{ mJ} \]

### Step 5: Calculate total energy (TE)
The total energy in simple harmonic motion is given by:
\[ TE = \frac{1}{2} m A^2 \omega^2 \]
Substituting the values:
\[ TE = \frac{1}{2} \times 0.01 \times \left(\frac{0.36}{\pi}\right)^2 \times \left(\frac{\pi}{3}\right)^2 \]
Calculating \( A^2 \) and \( \omega^2 \):
\[ A^2 = \left(\frac{0.36}{\pi}\right)^2 = \frac{0.1296}{\pi^2} \]
\[ \omega^2 = \left(\frac{\pi}{3}\right)^2 = \frac{\pi^2}{9} \]
Now substituting back:
\[ TE = \frac{1}{2} \times 0.01 \times \frac{0.1296}{\pi^2} \times \frac{\pi^2}{9} \]
\[ TE = \frac{0.01 \times 0.1296}{18} = \frac{0.001296}{18} = 0.000072 \text{ J} = 72 \text{ mJ} \]

### Step 6: Calculate potential energy (PE)
In simple harmonic motion, the potential energy can be calculated as:
\[ PE = TE - KE \]
Substituting the values:
\[ PE = 0.000072 - 0.000018 = 0.000054 \text{ J} = 54 \text{ mJ} \]

### Final Values
- Kinetic Energy (KE) = 18 mJ
- Potential Energy (PE) = 54 mJ
- Total Energy (TE) = 72 mJ