Question 156: Work, Energy and Power & Circular Motion

Question

A body of mass m starts moving from rest along x-axis so that its velocity varies as $\upsilon  = a\sqrt s $ where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

Accepted Answer

Given,
$\upsilon $ = a $\sqrt s $
$ \Rightarrow $$\,\,\,$ ${{ds} \over {dt}} = a\sqrt s $
$ \Rightarrow $$\,\,\,$ $\int\limits_0^t {{{ds} \over {\sqrt s }}}  = \int\limits_0^z {a\,dt} $
$ \Rightarrow $$\,\,\,$ 2$\sqrt s $ = at
$ \Rightarrow $$\,\,\,$ s = ${{{a^2}{t^2}} \over 4}$
= ${1 \over 2}.{{{a^2}} \over 2}.{t^2}$
$	herefore\,\,\,\,$ acceleration = ${{{a^2}} \over 2}$
$	herefore\,\,\,$ Force (F) = m $ 	imes $ ${{{a^2}} \over 2}$
$	herefore\,\,\,\,$ Work done = F. S
= ${{m{a^2}} \over 2} 	imes {{{a^2}{t^2}} \over 4}$
= ${{m{a^4}{t^2}} \over 8}$

Answer

${1 \over 8}\,$ m a
4
t
2

Answer

8 m a
4
t
2

Answer

4 m a
4
t
2

Answer

${1 \over 4}\,$ m a
4
t
2

Work, Energy and Power & Circular Motion

A body of mass m starts moving from rest along x-axis so that its velocity varies as \upsilon = a\sqrt s where a is a constant and s is the distance covered by the body. The total work done by all the forces acting on the body in the first t seconds after the start of the motion is :

Choose the correct answer: