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: \[ V_{{out}} = 7.2 \, {V}. \]
What is the first law of Kirchhoff of the electrical circuit? Find out the potential difference between the ends of 2 \(\Omega\) resistor with the help of Kirchhoff's law. See the figure:

Two projectile protons \( P_1 \) and \( P_2 \), both with spin up (along the \( +z \)-direction), are scattered from another fixed target proton \( T \) with spin up at rest in the \( xy \)-plane, as shown in the figure. They scatter one at a time. The nuclear interaction potential between both the projectiles and the target proton is \( \hat{\lambda} \vec{L} \cdot \vec{S} \), where \( \vec{L} \) is the orbital angular momentum of the system with respect to the target, \( \vec{S} \) is the spin angular momentum of the system, and \( \lambda \) is a negative constant in appropriate units. Which one of the following is correct?

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:
