Question

A block of mass m is connected to one end of a spring of spring constant k. The other end of the spring is fixed to a rigid support. Tile mass is released slowly so that the total energy of the system is then constituted by only the potential energy, then d is the maximum extension of the spring. Instead, if the mass is released suddenly from the same initial position, the maximum extension of the spring now is (g = acceleration due to gravity)

• A. $\frac{mg}{k}$
• B. $2d$
• C. $\frac{mg}{3k}$
• D. $4d$

Answer 1

Answer: (B)

$2d$

Answer 2

Case 1: If block is released slowly, Given, spring constant = kWhen mass m is suspended from spring, then d extension is developed in spring. From law of conservation of energy $mgh=\frac{1}{2}k{{d}^{2}}+mg(h-d)$ $mgh=\frac{1}{2}k{{d}^{2}}+mgh-mgd$ $mgd=\frac{1}{2}k{{d}^{2}}$ $\Rightarrow$ $d=\frac{2\,mg}{k}$ ?(i) $\Rightarrow$ $k=\frac{2mg}{d}$ Case 2: Again if block is released suddenly, further, then again, from law of conservation of energy $mgh=\frac{1}{2}k{{(d+y)}^{2}}+mg(h-d-y)$ $mg(d+y)=\frac{1}{2}k{{(d+y)}^{2}}$ $mg(d+y)=\frac{1}{2}\frac{2mg}{d}({{d}^{2}}+{{y}^{2}}+2dy)$ ${{d}^{2}}+dy={{d}^{2}}+{{y}^{2}}+2dy$ $\Rightarrow$ $y(y+d)=0$ $y=-d$ $\therefore$ Total displacement = (2d)