Two small spheres of mass 1 kg each are connected to each other by means of an unstretched spri

step 1 Hello and welcome to this video solution of numerate. So here we are given two small spheres of

Two small spheres of mass 1 kg each are connected to each...

Two small spheres of mass $1 \mathrm{~kg}$ each are connected to each other by means of an unstretched spring of force constant $k=100 \mathrm{~N} / \mathrm{m}$ and natural length $2.4 \mathrm{~m}$. The system is placed on a smooth horizontal surface and the two spheres are given velocities $v_{u}$, as shown in the figure. The maximum elongation in the spring is found to be $0.4 \mathrm{~m}$ during the subsequent motion. Calculate $v_{0}$ in $\mathrm{m} / \mathrm{s}$. Take $\sqrt{31}=5.6$

Key Concepts

Relative Motion and the Use of Reduced Mass

When analyzing two-body problems in spring-mass systems, it is useful to separate the motion into the center-of-mass motion and the relative motion. The concept of reduced mass simplifies the study of the internal oscillations, especially when the relative speed momentarily drops to zero at maximum spring extension, allowing for the calculation of energy stored in the spring.

Elastic Potential Energy in Springs

The energy stored in a spring when it is stretched or compressed is given by (1/2) k x², where k is the spring constant and x is the displacement from its natural length. This concept is essential for understanding how energy is stored and released in oscillatory motion.

Conservation of Momentum

In a frictionless system, the total momentum remains constant in the absence of external forces. For two objects connected by a spring, this principle helps determine the motion of the center of mass, which continues to move at a constant velocity even as the objects oscillate relative to each other.

Conservation of Energy

Energy conservation is crucial in spring–mass systems. Initially, all energy is kinetic, and as the spring stretches, some kinetic energy is converted into elastic potential energy. At maximum spring extension, the relative kinetic energy between the masses is momentarily zero, and the maximum elastic potential energy is stored in the spring.

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