Question

A mass m, lying on a horizontal, frictionless surface, is connected to one end of a spring of spring constant $k$. The other end of the spring is connected to a wall, as shown in the figure. At $t=0$, the mass is given an impulse $J$ as shown. Taking left direction as positive, the time dependence of the displacement, acceleration and the velocity of the mass are:

• A. $x=\frac{J}{\sqrt{mk}}sin\left(\sqrt{\frac{k}{m}}t\right)$
• B. $v=\frac{J}{m}sin(\sqrt{\frac{k}{m}}t+\frac{3\pi }{2})$
• C. $a=\frac{J\sqrt{k}}{\sqrt{m^3}}sin\left(\sqrt{\frac{k}{m}}t\right)$
• D. $a=\frac{J\sqrt{k}}{\sqrt{m^3}}sin(\sqrt{\frac{k}{m}}t+\pi )$

Answer 1

Answer: (A), (D)

$a=\frac{J\sqrt{k}}{\sqrt{m^3}}sin(\sqrt{\frac{k}{m}}t+\pi )$

$v=\frac{J}{m}=A\omega =A\sqrt{\frac{k}{m}}$
$A=\frac{J}{\sqrt{mk}}$
As the mass starts from mean position,
$x=\frac{J}{\sqrt{mk}}sin\left(\sqrt{\frac{k}{m}}t\right)$

$v=\frac{dx}{dt}=\frac{J}{\sqrt{mk}}\times \sqrt{\frac{k}{m}}cos\left(\sqrt{\frac{k}{m}}t\right)$
$v=\frac{J}{m}cos\left(\sqrt{\frac{k}{m}}t\right)$

$a=-{\omega }^{2}x=-\frac{k}{m}\times \frac{J}{\sqrt{mk}}sin\left(\sqrt{\frac{k}{m}}t\right)=\frac{J\sqrt{k}}{\sqrt{m^3}}sin\left(\sqrt{\frac{k}{m}}t\right)$
$a=\frac{J\sqrt{k}}{\sqrt{m^3}}sin(\sqrt{\frac{k}{m}}t+\pi )$