Question

A system consists of two identical cubes each of mass m linked together by a mass less spring of spring constant K. The spring is compressed by x connecting cubes by thread. Find minimum value of x for which lower cube will bounce up after the thread has been burnt.

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

Answer 1

Answer: (B)

$\frac { 3 \mathrm { mg } } { \mathrm { k } }$

Answer 2

The elongation produced x' be such that kx' = mg. Apply energy conservation as at the maximum elongation u_block = 0.

$\frac { 1 } { 2 }$ kx² = mg (x + x')+ $\frac { 1 } { 2 }$ kx'²

x² - $\frac { 2 \mathrm { mgx } } { \mathrm { k } } - 2 \frac { \mathrm { mg } } { \mathrm { k } }$ x' – x'²

or x² - $\frac { 2 \mathrm { mg } } { \mathrm { k } } \mathrm { x } - \frac { 2 \mathrm { mg } } { \mathrm { k } } \left( \frac { \mathrm { mg } } { \mathrm { k } } \right) - \left( \frac { \mathrm { mg } } { \mathrm { k } } \right) ^ { 2 } = 0$

or x = $\frac { 3 \mathrm { mg } } { \mathrm { k } }$