Question

A particle of mass $10gm$ moves in a field where potential energy per unit mass is given by expression $v=8\times {10}^4{x}^{2}erg/gm.$ If the total energy of the particle is $8\times {10}^{7}erg$ then the relation between $x$ and $t$ is:
[$\varphi =constant$]

• A. $x=10\sin (400t+\varphi )cm$
• B. $x=\sin (400t+\varphi )m$
• C. $x=10\sin (4t+\varphi )m$
• D. $x=100\sin (4t+\varphi )m$

Answer 1

The correct option is C $x=10\sin (4t+\varphi )m$

$V=8\times {10}^4{x}^{2}erg,mass=10gm,TE=8\times {10}^{7}erg$

$\therefore$ Potential energy $=8\times {10}^4{x}^2\times 10\times {10}^{-7}J$

$=8x^2\times {10}^{-2}J$

$\begin{array}{c}TE=8\times {10}^7\times {10}^{-7}J\\ =8J\\ F=\left|\frac{dy}{dx}\right|=16x\times {10}^{-2}N\\ \Rightarrow k=16\times {10}^{-2}\\ \Rightarrow k{A}^2=8\\ \Rightarrow \frac{1}{2}\times 16\times {10}^{-2}\times x^2=8\\ \Rightarrow A=10\\ W=\sqrt{\frac{K}{M}}=\sqrt{\frac{16\times {10}^{-2}}{10\times {10}^{-3}}}=4\\ \therefore x=10\left(\sin 4t+\varphi \right)m\end{array}$

$\therefore$ Option $C$ is correct.