Question

A block of a mass $mkg$ moving with kinetic energy $3J$ on a smooth horizontal surface hits the free end of a spring, whose other end is attached to a rigid wall as shown in the figure. If restoring force applied by the spring is given by $f\left(x\right)=-2x-x^3$ ($N$). Then the maximum compression in the spring approximately

• A. $2.20m$
• B. $1.51m$
• C. $1.41m$
• D. $1.67m$

Answer 1

Answer: (C)

$1.41m$

Given KE= 3J.

let $E_i$ is initial energy and $E_f$ be final energy.

$E_i=K{E}_i+P{E}_i$

$E_i=3J+0=3J$

at maximum compression, block will stop for a moment so $K{E}_f=0$

$E_f=K{E}_f+P{E}_f$

$E_f=0+P{E}_f$

Total energy remains conserved

$E_i=E_f$

$3J=P{E}_f$

Energy by Force $fx$ is given by $-\int f\left(x\right)dx$

$PE\left(x\right)=-\int f\left(x\right)dx$

$PE\left(x\right)=-(-2x-x^3)dx$

$PE\left(x\right)=x^2+\frac{x^4}{4}$

from above result $P{E}_f=$ $PE\left(x_{max}\right)=x^2+\frac{x^4}{4}$

$3=x^2+\frac{x^4}{4}$

$x^2=2$

$x=\sqrt{2}=1.41m$