Sinusoidal waveform conversion: A sinusoidal voltage has a peak-to-peak value of 100 V. Determine its root-mean-square (RMS) value, keeping the basic AC relationships explicit.

Introduction / Context:RMS values are essential in AC circuit analysis because they relate sinusoidal voltages and currents to equivalent DC heating effects. Given a peak-to-peak voltage, we often need to compute RMS to size components and evaluate power accurately.

Given Data / Assumptions:

• Peak-to-peak voltage V_pp = 100 V.
• Pure sinusoidal waveform.
• Standard RMS relation for sinusoids applies.

Concept / Approach:

For a sinusoid, V_pp = 2 * V_m, where V_m is the positive peak (amplitude). The RMS value is V_rms = V_m / √2. Therefore, convert V_pp to V_m, then divide by √2 to obtain the RMS value.

Step-by-Step Solution:

Verification / Alternative check:

Cross-check: If V_rms = 35.35 V, then V_m = V_rms * √2 ≈ 50 V and V_pp = 2 * V_m = 100 V, matching the given peak-to-peak value.

Why Other Options Are Wrong:

• 50 V: That is the peak, not the RMS.
• 70.7 V: That would be RMS if the peak were 100 V, but here 100 V is peak-to-peak.
• 141.41 V: That is 2 * 100/√2, not applicable here.
• 25 V: No standard relation yields this value from 100 V peak-to-peak.

Common Pitfalls:

• Confusing V_pp with V_m; always halve V_pp to get the amplitude before converting to RMS.

Final Answer:

35.35 V