Question

Find to what height $h$ the centre of gravity of this system will rise if the initial compression of the spring $\Delta l=\frac{7mg}{\kappa }$.

• A. $\frac{mg}{\kappa }$
• B. $\frac{4mg}{\kappa }$
• C. $\frac{8mg}{\kappa }$
• D. $\frac{2mg}{\kappa }$

Answer 1

The correct option is C $\frac{8mg}{\kappa }$

Let v be the velocity of the upper cube when the lower cube breaks off the floor, then :

$\frac{1}{2}m{v}^2=\frac{1}{2}k(\Delta l^2-l^2)-mg(l+\Delta l)......\left(1\right)wherel=\frac{mg}{k}and\Delta l=7\frac{mg}{k}{v}^2=32\frac{m{g}^2}{k}......\left(2\right)atthispoint{v}_{cm}=\frac{mv+0}{2m}=\frac{v}{2}LettheCOMofthesystemrisesby\Delta y,thenfromenergyconservation:\frac{1}{2}\times 2m{v}_{cm}^2=2m\times g\times \Delta y\Rightarrow \Delta y=\frac{4mg}{k}buttillthebreakoffpointtheCOMhasalreadyelevatedby\frac{m(l+\Delta l)+0}{2m}=\frac{4mg}{k}totalelevation=\frac{8mg}{k}$