Question 21: Work, Energy and Power & Circular Motion

Question

Potential energy ( $V$ ) versus distance ( $x$ ) is given by the graph. Rank various regions as per the magnitudes of the force ( $F$ ) acting on a particle from high to low.

Accepted Answer

The force (F) acting on a particle is defined as the negative gradient of its potential energy (V) with respect to position (x) :
$$ \mathrm{F}=-\frac{\mathrm{dV}}{\mathrm{dx}} $$
Here, $\frac{d V}{d x}$ is the slope of the $V$ versus $x$ graph. So, the magnitude of the force $(|F|)$ is the absolute value of the slope in each region :
$$ |\mathrm{F}|=\mid 	ext { slope of the } \mathrm{V}-\mathrm{x} 	ext { graph } \mid $$
Using the formula of slope
$$ 	ext { Slope }=m=\frac{V_2-V_1}{x_2-x_1} $$
For region AB (from $\mathrm{x}=0$ to $\mathrm{x}=1$ ) :
Points are ( 0,0 ) and ( 1,1 )
$$ \left|\mathrm{F}_{\mathrm{AB}}ight|=\left|\frac{1-0}{1-0}ight|=1 	ext { unit } $$
For region BC (from $\mathrm{x}=1$ to $\mathrm{x}=2$ ) :
Points are $(1,1)$ and $(2,4)$
$$ \left|\mathrm{F}_{\mathrm{BC}}ight|=\left|\frac{4-1}{2-1}ight|=3 	ext { unit } $$
For region CD (from $\mathrm{x}=2$ to $\mathrm{x}=4$ ) :
Points are (2,4) and (4,4)
$$ \left|\mathrm{F}_{\mathrm{CD}}ight|=\left|\frac{4-4}{4-2}ight|=0 	ext { unit } $$
For region DE (from $\mathrm{x}=4$ to $\mathrm{x}=6$ ):
Points are $(4,4)$ and $(6,3)$
$$ \left|\mathrm{F}_{\mathrm{DE}}ight|=\left|\frac{3-4}{6-4}ight|=0.5 	ext { unit } $$
The descending order of the magnitude of forces is,
$$ 3>1>0.5>0 \Rightarrow \mathrm{~F}_{\mathrm{BC}}>\mathrm{F}_{\mathrm{AB}}>\mathrm{F}_{\mathrm{DE}}>\mathrm{F}_{\mathrm{CD}} $$
Therefore, region $B C$ has the greatest slope and thus the strongest force, while region CD has no slope and zero force.
Hence, the correct option is (A).

Answer

$F_{B C}>F_{A B}>F_{D E}>F_{C D}$

Answer

$F_{B C}>F_{C D}>F_{D E}>F_{A B}$

Answer

$F_{C D}>F_{A B}>F_{B C}>F_{D E}$

Answer

$F_{C D}>F_{D E}>F_{A B}>F_{B C}$

Work, Energy and Power & Circular Motion

Potential energy ( V ) versus distance ( x ) is given by the graph. Rank various regions as per the magnitudes of the force ( F ) acting on a particle from high to low.

Choose the correct answer: