Question

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 unstretched. The maximum extension in the spring is

• A. $\frac{4mg}{k}$
• B. $\frac{2mg}{k}$
• C. $\frac{mg}{k}$
• D. $\frac{mg}{2k}$

Answer 1

The correct option is B $\frac{2mg}{k}$

Let the spring extend in length $y$ when mass $m$ is hung freely at its lower end.
The elastic potential energy stored in the spring $=\frac{1}{2}k{y}^2$
when the mass is released, the gravitational potential energy lost is $=mgy$
From energy conservation principle,
Increase in elastic
PE = Decrease in Gravitational PE
$=\frac{1}{2}k{y}^2=mgy\Rightarrow y=\frac{2mg}{k}$
Hence, the correct answer is option (b).