The figure shows an opamp circuit with a 5.1 V Zener diode in the feedback loop. The opamp runs from \( \pm 15 \, {V} \) supplies. If a \( +1 \, {V} \) signal is applied at the input, the output voltage (rounded off to one decimal place) is:
Step 1: In the given op-amp circuit, the Zener diode is in the feedback loop with a breakdown voltage of 5.1 V. The Zener diode operates in the breakdown region, which means the voltage across it will be clamped to 5.1 V, regardless of the output voltage of the op-amp.
Step 2: The op-amp operates in the linear region, and the voltage across the Zener diode remains constant at 5.1 V. The input voltage is \( +1 \, {V} \), and since the op-amp is configured in a feedback loop, the output voltage will be adjusted to maintain the voltage across the diode at 5.1 V.
Step 3: Since the Zener diode has a voltage of 5.1 V, the output voltage will be clipped at \( +5.1 \, {V} \). Therefore, the output voltage of the op-amp is: \[ V_{{out}} = 7.2 \, {V}. \]
Two batteries of emf's \(3V \& 6V\) and internal resistances 0.2 Ω \(\&\) 0.4 Ω are connected in parallel. This combination is connected to a 4 Ω resistor. Find:
(i) the equivalent emf of the combination
(ii) the equivalent internal resistance of the combination
(iii) the current drawn from the combination
A wheel of mass \( 4M \) and radius \( R \) is made of a thin uniform distribution of mass \( 3M \) at the rim and a point mass \( M \) at the center. The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length \( 2R \), with one end fixed at \( O \), as shown in the figure. The wheel rolls without slipping on horizontal ground with angular speed \( \Omega \). If \( \vec{L} \) is the total angular momentum of the wheel about \( O \), then the magnitude \( \left| \frac{d\vec{L}}{dt} \right| = N(MR^2 \Omega^2) \). The value of \( N \) (in integer) is: