The average value of current `i=I_(m) sin omega t from t=(pi)/(2 omega )` to `t=(3 pi)/(2 omega)` si how many times of `(I_m)`?

Explanation

To find the average value of the current \( i = I_m \sin(\omega t) \) from \( t = \frac{\pi}{2\omega} \) to \( t = \frac{3\pi}{2\omega} \), we will follow these steps:

### Step 1: Set Up the Average Value Formula
The average value of a function \( f(t) \) over an interval \([t_1, t_2]\) is given by:
\[ I_{\text{avg}} = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} f(t) \, dt \]
In our case, \( f(t) = I_m \sin(\omega t) \), \( t_1 = \frac{\pi}{2\omega} \), and \( t_2 = \frac{3\pi}{2\omega} \).

### Step 2: Calculate the Time Interval
Calculate the time interval:
\[ t_2 - t_1 = \frac{3\pi}{2\omega} - \frac{\pi}{2\omega} = \frac{2\pi}{2\omega} = \frac{\pi}{\omega} \]

### Step 3: Set Up the Integral
Now we set up the integral:
\[ I_{\text{avg}} = \frac{1}{\frac{\pi}{\omega}} \int_{\frac{\pi}{2\omega}}^{\frac{3\pi}{2\omega}} I_m \sin(\omega t) \, dt \]
This simplifies to:
\[ I_{\text{avg}} = \frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}}^{\frac{3\pi}{2\omega}} I_m \sin(\omega t) \, dt \]

### Step 4: Factor Out Constants
Since \( I_m \) is a constant, we can factor it out of the integral:
\[ I_{\text{avg}} = \frac{I_m \omega}{\pi} \int_{\frac{\pi}{2\omega}}^{\frac{3\pi}{2\omega}} \sin(\omega t) \, dt \]

### Step 5: Evaluate the Integral
To evaluate the integral \( \int \sin(\omega t) \, dt \):
\[ \int \sin(\omega t) \, dt = -\frac{1}{\omega} \cos(\omega t) \]
Now we evaluate it from \( t = \frac{\pi}{2\omega} \) to \( t = \frac{3\pi}{2\omega} \):
\[ \int_{\frac{\pi}{2\omega}}^{\frac{3\pi}{2\omega}} \sin(\omega t) \, dt = \left[-\frac{1}{\omega} \cos(\omega t) \right]_{\frac{\pi}{2\omega}}^{\frac{3\pi}{2\omega}} \]
Calculating the limits:
\[ = -\frac{1}{\omega} \left( \cos\left( \frac{3\pi}{2} \right) - \cos\left( \frac{\pi}{2} \right) \right) = -\frac{1}{\omega} \left( 0 - 0 \right) = 0 \]

### Step 6: Substitute Back into the Average Value Formula
Substituting back into the average value formula:
\[ I_{\text{avg}} = \frac{I_m \omega}{\pi} \cdot 0 = 0 \]

### Step 7: Conclusion
Thus, the average value of the current \( I_{\text{avg}} \) is \( 0 \). Since the question asks how many times \( I_m \) this average value is, we conclude:
\[ \text{The average value is } 0 \text{ times } I_m. \]