If the extension of the spring is x0 at time t, then the...

If the extension of the spring is $x_{0}$ at time $t$, then the displacement of the second block at this instant is (a) $\frac{F l^{2}}{2 m}-x_{0}$ (b) $\frac{1}{2}\left(\frac{F l^{2}}{2 m}+x_{0}\right)$ (c) $\frac{1}{2}\left(\frac{2 F t^{2}}{m} \quad x_{o}\right)$ (d) $\frac{1}{2}\left(\frac{F t^{2}}{2 m} \quad x_{0}\right)$

If the extension of the spring is $x_{0}$ at time $t$, then the displacement of the second block at this instant is
(a) $\frac{F l^{2}}{2 m}-x_{0}$
(b) $\frac{1}{2}\left(\frac{F l^{2}}{2 m}+x_{0}\right)$
(c) $\frac{1}{2}\left(\frac{2 F t^{2}}{m} \quad x_{o}\right)$
(d) $\frac{1}{2}\left(\frac{F t^{2}}{2 m} \quad x_{0}\right)$

Key Concepts

Relative Displacement and Constraint Relationships

When objects are connected by a mechanism such as a spring, the displacement of each object is not independent but is related through the constraints of the connection. Understanding these relative displacement relationships is key to determining the motion of each component in the system, as the extension or compression of the spring directly ties the positions of the connected masses.

Coupled Mass-Spring Systems

In coupled mass-spring systems, two or more masses are connected by springs, which introduce forces that depend on their extension or compression. Analyzing these systems involves understanding both the individual motions of the masses and the constraint relationships imposed by the spring, often using Hooke’s Law to relate the spring force to its displacement.

Newton's Second Law

Newton's Second Law is fundamental in dynamics, stating that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle serves as a cornerstone for analyzing how forces affect the motion of individual or multiple connected masses in a system.

Kinematics of Uniform Acceleration

Kinematic equations describe the motion of an object under constant acceleration, relating displacement, velocity, acceleration, and time. These equations allow us to calculate how far an object has moved over a period in scenarios where acceleration is uniform, and they are essential for solving displacement problems in dynamics.

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