An ideal spring with spring constant k is hung from the...

An ideal spring with spring constant $k$ is hung from the ceiling and a block of mass $m$ is attached to its lower end. The mass is released with the spring initially unstreched. The maximum extension in the spring is (a) $\frac{4 m g}{k}$ (b) $\frac{2 m g}{k}$ (c) $\frac{m g}{k}$ (d) $\frac{m g}{2 k}$

An ideal spring with spring constant $k$ is hung from the ceiling and a block of mass $m$ is attached to its lower end. The mass is released with the spring initially unstreched. The maximum extension in the spring is
(a) $\frac{4 m g}{k}$
(b) $\frac{2 m g}{k}$
(c) $\frac{m g}{k}$
(d) $\frac{m g}{2 k}$

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Key Concepts

Hooke’s Law

Hooke’s Law states that the force exerted by a spring is proportional to its displacement from its natural length, expressed mathematically as F = -kx. This law is fundamental in understanding how springs behave under applied forces, whether in static or dynamic situations.

Elastic Potential Energy

Elastic potential energy is the energy stored in a spring when it is deformed (stretched or compressed). It is given by the formula U = ½ kx², where k is the spring constant and x is the displacement. This concept is essential for analyzing energy conversions in systems involving springs.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, typically calculated as U = mgh for a vertical displacement h. This concept is important in problems where objects move under the influence of gravity, such as a mass attached to a spring.

Conservation of Mechanical Energy

The conservation of mechanical energy principle states that in the absence of non-conservative forces (like friction), the total mechanical energy (the sum of kinetic, gravitational potential, and elastic potential energies) remains constant. This principle is used to analyze the motion of systems like the mass-spring oscillator when energy is transformed between different forms.

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