Q. A single conservative force $F(x)$ acts on a particle that moves along the $x$-axis. The graph of the potential energy with $x$ is given. At $x=5\, m$, the particle has a kinetic energy of $50 \,J$ and its potential energy is related to position ' $x$ ' as $U=15+(x-3)^{2}$ Joule, where $x$ is in meter. Then :

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A

The mechanical energy of system is 69

B

The mechanical energy of system is 19 J

C

At $x=3$, the kinetic energy of particle is minimum

D

The maximum value of kinetic energy is 54 J

Solution:

$U =15+(x-3)^{2} $
$U(5) =19 \, \&\, K E(5)=50$
Total mechanical energy $=50+19=69 \,J$
$U_{\text {min. }} \equiv U(x-3) =15 \,J $
$K E_{\max .} =69-U_{\min .}=69-15=54 \,J$

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The mechanical energy of system is 69

The mechanical energy of system is 19 J

At $x=3$, the kinetic energy of particle is minimum

The maximum value of kinetic energy is 54 J

$U =15+(x-3)^{2} $ $U(5) =19 \, \&\, K E(5)=50$ Total mechanical energy $=50+19=69 \,J$ $U_{\text {min. }} \equiv U(x-3) =15 \,J $ $K E_{\max .} =69-U_{\min .}=69-15=54 \,J$

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4. A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_{c}$ is varying with time $t$ as $a_{c}=k^{2} \, rt^{2}$ , where $k$ is a constant. The power delivered to the particle by the force acting on it is

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$U =15+(x-3)^{2} $
$U(5) =19 \, \&\, K E(5)=50$
Total mechanical energy $=50+19=69 \,J$
$U_{\text {min. }} \equiv U(x-3) =15 \,J $
$K E_{\max .} =69-U_{\min .}=69-15=54 \,J$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ?
(density of mercury $= 13600 kg/m^{3})$