Question

Hi Actually I'm facing a problem that I knoe all the values but whenever I'm going to solve it seems that I'm unable to find out the answer. Can you help me out?

Here, values :-

$T=10\ N$
$m_0=10^{-2}\ kg\,m^{-1}$
$a=9\times10^{-3}\ kg\,m^{-2}$

When :+

$\longrightarrow \dfrac{dx}{dt}=\sqrt{\dfrac{T}{m_0+ax}}$

Note : solve this using integrate! ​

Answer 1

$\Large{\underbrace{\sf{\purple{Required\:Answer:}}}}$

Here,we are given :-

• $\sf \dfrac{dx}{dt}=\sqrt{\dfrac{T}{m_0+ax}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀

$\sf{\implies \red { \dfrac{dx}{dt}=\sqrt{\dfrac{T}{m_0+ax}}}}$

$\sf{\implies dt=\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

Integrating:-

$\displaystyle\sf{\implies\int\limits_0^tdt=\int\limits_0^{10}\sqrt{\dfrac{m_0+ax}{T}}\ dx}$

$\displaystyle\sf{\implies t=\dfrac{1}{a\sqrt T}\cdot\dfrac{2}{3}\left[(m_0+ax)^{\frac{3}{2}}\right]_0^{10}}$

Now, we are given some values.

• $\sf{T=10\ N}$
• $\sf{m_0=10^{-2}\ kg\,m^{-1}}$
• $\sf{a=9\times10^{-3}\ kg\,m^{-2}}$

_____________________________________

• So, we have to put the above values for getting the proper solution,.Let's do this!

$\displaystyle\sf{\implies t=\dfrac{1}{9\times10^{-3}\sqrt{100}}\cdot\dfrac{2}{3}\left[\left(10^{-2}+(9\times10^{-3})x\right)^{\frac{3}{2}}\right]_0^{10}}$

$\displaystyle\sf{\implies t=\dfrac{200}{27}\left(10^{-\dfrac{3}{2}}-10^{-3}\right)}$

$\displaystyle\sf{\implies\red{\underline{\underline{t=0.2268\ s}}}}$