In Fig. 4.20 shown here, pulley and spring are ideal. If k...

In Fig. $4.20$ shown here, pulley and spring are ideal. If $k$ is spring constant of spring, the potential energy stored in it is $\left(m_{1}>m_{2}\right)$ Fig. $4.20$ (A) $\frac{2 m_{1}^{2} g^{2}}{k}$ (B) $\frac{2 m_{2}^{2} g^{2}}{k}$ (C) $\frac{\left(m_{1}+m_{2}\right)^{2} g^{2}}{k}$ (D) $\frac{1}{2} \frac{\left(m_{1}-m_{2}\right)^{2} g^{2}}{k}$

In Fig. $4.20$ shown here, pulley and spring are ideal. If $k$ is spring constant of spring, the potential energy stored in it is $\left(m_{1}>m_{2}\right)$
Fig. $4.20$
(A) $\frac{2 m_{1}^{2} g^{2}}{k}$
(B) $\frac{2 m_{2}^{2} g^{2}}{k}$
(C) $\frac{\left(m_{1}+m_{2}\right)^{2} g^{2}}{k}$
(D) $\frac{1}{2} \frac{\left(m_{1}-m_{2}\right)^{2} g^{2}}{k}$

Key Concepts

Force Balance in Pulley Systems

In systems with an ideal pulley, the tension in the rope is the same on both sides of the pulley, allowing the gravitational forces on the masses to be directly related to the spring force. The difference in weight between the masses provides the net force that stretches the spring until equilibrium is reached. Understanding this balance is key to analyzing the system and predicting its behavior under the influence of gravity and spring forces.

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from its equilibrium position, mathematically described by F = kx, where k is the spring constant and x is the displacement. This principle is fundamental in analyzing systems involving springs as it allows determining how much a spring stretches or compresses under a given force.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is either stretched or compressed. The energy is given by the formula U = 1/2 kx², where k is the spring constant and x is the displacement from the equilibrium position. This concept is crucial in understanding energy transformations in mechanical systems that include springs.

Equilibrium Condition in a Spring-Mass System

In a spring-mass system connected to different masses, equilibrium is achieved when the net force acting on the system is zero. For an ideal pulley and spring system, establishing equilibrium involves balancing the gravitational forces acting on the masses with the restoring force of the spring. This balance determines how much the spring stretches, which in turn is used to calculate the elastic potential energy stored in the spring.

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