A charged 10 micro farad capacitor is connected to a 81 mH inductor. What is the angular frequency of free oscillations of the circuit?

Answer

To find the angular frequency of free oscillations in an LC circuit (which consists of an inductor and a capacitor), we can use the formula for the angular frequency (ω) given by:

\[ \omega = \frac{1}{\sqrt{LC}} \]

where:
- \(L\) is the inductance in henries (H),
- \(C\) is the capacitance in farads (F).

### Step 1: Convert the given values to standard units
- The capacitance \(C\) is given as \(10 \, \mu F\), which is \(10 \times 10^{-6} \, F\).
- The inductance \(L\) is given as \(81 \, mH\), which is \(81 \times 10^{-3} \, H\).

### Step 2: Substitute the values into the formula
Now we can substitute the values of \(L\) and \(C\) into the formula for angular frequency:

\[ \omega = \frac{1}{\sqrt{(81 \times 10^{-3}) \times (10 \times 10^{-6})}} \]

### Step 3: Calculate the product \(LC\)
First, calculate \(LC\):

\[ LC = (81 \times 10^{-3}) \times (10 \times 10^{-6}) = 81 \times 10^{-9} = 8.1 \times 10^{-8} \, H \cdot F \]

### Step 4: Calculate the square root
Now, take the square root of \(LC\):

\[ \sqrt{LC} = \sqrt{8.1 \times 10^{-8}} = \sqrt{8.1} \times 10^{-4} \approx 2.84 \times 10^{-4} \, s \]

### Step 5: Calculate the angular frequency
Now, substitute back into the formula for \(\omega\):

\[ \omega = \frac{1}{\sqrt{LC}} = \frac{1}{2.84 \times 10^{-4}} \approx 3520.4 \, rad/s \]

### Final Answer
Thus, the angular frequency of free oscillations of the circuit is approximately:

\[ \omega \approx 3520.4 \, rad/s \]

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