Question

A body of mass $1kg$ falls freely from a height of $100m$ on a platform of mass$3kg$ which is mounted on a spring having spring constant $k=1.25\times {10}^{6}N/m$. The body sticks to the platform and the spring's maximum compression is found to be$x$. Given that $g=10m{s}^{–2}$, the value of $x$will be close to:

• A. $4cm$
• B. $8cm$
• C. $80cm$
• D. $40cm$

Answer 1

The correct option is A

$4cm$

Step 1. Given Data:

Mass of body, $m=1kg$

Height from which body falls freely, $h=100m$

Mass of platform, $M=3kg$

Spring Constant, $k=1.25\times {10}^{6}N/m$

Maximum compression is $x$

Acceleration due to gravity, $g=10m{s}^{–2}$

Step 2. Solving for maximum compression$x$

Now, when a body of any mass falls from height then its energy stored is in the form of potential energy which can be given as,

$P=mgh$

Where, $m$ is mass, $h$ is height and $g$ is gravitational acceleration.

Considering the potential energy of mass $1kg$, for height of $100m$ can be given as,

$m_1=1kg$

$h_1=100m$

$P_1=m_{1}g{h}_1$ ….(1)

Now, when the body of mass $1kg$ falls on the platform of mass $3kg$ is attached to spring, the spring compresses till $x$ height, so the potential energy stored due to compression can be given as,

$P_2=(m_1+m_2)gx$ ….(2)

Now, this potential energy gets converted into kinetic energy of spring which can be seen mathematically as,

$KE=\frac{1}{2}k{x}^2$ ….(3)

Where, $k$ is spring constant and $x$ is displacement of spring due to compression.

Now, equating expression (1), (2) and (3) we will get,

$P_1+P_2=KE$

$\Rightarrow$$m_{1}g{h}_1+(m_1+m_2)gx=\frac{1}{2}k{x}^2$

On substituting the values in the expression, we will get

$\Rightarrow$$\left(1\right)\left(10\right)\left(100\right)+(1+3)\left(10\right)x=\frac{1}{2}1.25\times {10}^6{x}^2$

$\Rightarrow$ $1000+40x=625000x^2$

$\Rightarrow$ $625000x^2-40x-1000=0$

$\Rightarrow$ $x=\frac{-(-40)\pm \sqrt{{(-40)}^2-4\left(625000\right)(-1000)}}{2\left(62500\right)}$

Solving the expression, the value of $x$ can be given as,

$\Rightarrow$$x=0.04morx=-0.039m$

$\Rightarrow$$x=4cmorx=-3.9cm\simeq -4cm$

$\Rightarrow$$x=4cm$

Hence, option A is correct.