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

The Correct Option is B

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)