Question 8: Work, Energy and Power & Circular Motion

Question

Given below are two statements :
Statement I : An object moves from position $r_1$ to position $r_2$ under a conservative force field $\vec{F}$. The work done by the force is $W=-\int\limits_{r_1}^{r_2} \vec{F} \cdot \overrightarrow{d r}$.
Statement II : Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

Let a conservative force $\overrightarrow{\mathrm{F}}$ is acting on an object. For infinitesimally small displacement $\mathrm{d} \overrightarrow{\mathrm{r}}=\mathrm{dx} \hat{\mathrm{i}}+\mathrm{dy} \hat{\mathrm{j}}+\mathrm{dz} \hat{\mathrm{k}}$, the work done is,
$$ \mathrm{dW}=\overrightarrow{\mathrm{F}} \cdot \mathrm{~d} \overrightarrow{\mathrm{r}} $$
The work done (W) by a variable force $\vec{F}$ when displacing an object from an initial position vector $\vec{r}_1$ to a final position vector $\overrightarrow{\mathrm{r}}_2$ is the total work done in moving the object,
The work done is, $W=\int\limits_{r_1}^{r_2} \vec{F} \cdot d \vec{r}$
So, this formula is valid for all forces whether it is conservative or non-conservative force. But the statement I states that the work done is $W=-\int_{r_1}^{r_2} \vec{F}$. $d \vec{r}$. Hence, the statement $I$ is false.
Work done by a conservative force is path independent. And the work done by the conservative force is negative of the change in potential energy, i.e., the work done by the conservative force is stored in object as potential energy.
The work done depends only on the initial and final positions.
$$ \mathrm{W}=\int\limits_{\overrightarrow{\mathrm{r}}_1}^{\overrightarrow{\mathrm{r}}_2} \overrightarrow{\mathrm{~F}} \cdot \mathrm{~d} \overrightarrow{\mathrm{r}}=-\left(\mathrm{U}_{\mathrm{f}}-\mathrm{U}_{\mathrm{i}}\right) $$
For a particle going from point A to point B , there are infinitely many possible paths. So, it is a geometrical fact and is correct.
A force is called conservative if $\oint \overrightarrow{\mathrm{F}} \cdot \mathrm{d} \overrightarrow{\mathrm{r}}=0$
That means work done in a closed path is zero.
Hence for motion from A to B :
$$ \mathrm{W}_{\mathrm{A} \rightarrow \mathrm{~B}}=\mathrm{U}_{\mathrm{A}}-\mathrm{U}_{\mathrm{B}} $$
So work done depends only on initial position and final position and not on the path followed.
But the statement says, "work done changes with path for a conservative force", so the statement is incorrect because for conservative forces, straight path, curved path, zig-zag path all give the same work between the same two points. Hence, the statement II is incorrect.
Therefore, both Statement I and Statement II are false. Hence, the correct option is (C).

Answer

Statement I is true but Statement II is false

Answer

Statement I is false but Statement II is true

Answer

Both Statement I and Statement II are false

Answer

Both Statement I and Statement II are true

Work, Energy and Power & Circular Motion

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