Two fixed point particles, each of charge $+q$, are separated by a distance $2 L$. Another point charge $+q$ of mass $m$ is oscillating about its equilibrium position as indicated in the figure below. The time period of oscillation is given by $T=$ $2 \pi^{3 / 2} \alpha \sqrt{m} / q$. Given that $\epsilon_0$ is the

permittivity of free space, which of the following options is the dimensionally correct expression for $\alpha$ in SI units?

$$ \text { What is the effective resistance between } A \text { and } B \text { in the circuit shown below? } $$

57. Consider a Carnot heat engine operating between a heat reservoir at temperature 600 K , and an external atmosphere at temperature 300 K . In one cycle, 1000 kJ of heat is extracted from the heat reservoir, and the associated work is input to a reversible refrigerator that operates between 200 K and the same external atmosphere at 300 K . The refrigerator completes one cycle and releases heat into the atmosphere. How much heat is released into the atmosphere at the end of one cycle of the combined system?

Two spherical bodies A and B each of mass $M$ and radius $R$ are located such that their centers are apart by a distance of $4 R$ as shown in the figure. An object $C$ of mass $m$ is thrown from the surface of A directly towards the center of B with a speed $v_0=2 v_{\text {min }}$, where $v_m$ in is the minimum speed needed for $C$ to reach the surface of $B$. Given that $G$ is the universal gravitational constant, how does the speed $v(x)$ of C change as a function of its distance from the center of A?