Conservative and Non-Conservative forces and Potential Energy Questions in English

(A) The relationship between a conservative force $F$ and the potential energy $U$ is given by $F = -\frac{dU}{dr}$.
Work done by a conservative force is defined as $W = -\Delta U = -(U_f - U_i) = U_i - U_f$.
If the work done $W$ is positive,then $U_i - U_f > 0$,which implies $U_i > U_f$.
Therefore,the potential energy of the body decreases.

(D) Potential energy is a property of a system in a conservative force field.
$1$. It is defined only in conservative fields because the work done by conservative forces is path-independent.
$2$. The change in potential energy $\Delta U$ is defined as the negative of the work done by conservative forces: $\Delta U = -W_c$.
$3$. It can also be defined as the work done by an external agent to move a system from one configuration to another without any change in kinetic energy $(\Delta K = 0)$,which equals the change in potential energy.
Therefore,all the given statements are correct. The correct option is $D$.

(D) The relationship between force $F$ and potential energy $U$ is given by $F = -\frac{dU}{dx}$.
This means the force is the negative of the slope of the $U-x$ graph.
$1$. For the interval $0 \rightarrow x_1$: The slope of the $U-x$ graph is positive and constant. Therefore, the force $F = -(\text{positive constant})$ is negative and constant.
$2$. For the interval $x_1 \rightarrow x_2$: The potential energy $U$ is constant, so the slope is zero. Therefore, the force $F = 0$.
$3$. For the interval $x_2 \rightarrow x_3$: The slope of the $U-x$ graph is negative and constant. Therefore, the force $F = -(\text{negative constant})$ is positive and constant.
Comparing this with the given options, the graph in option $D$ correctly represents a negative constant force, followed by zero force, and then a positive constant force.

(B) For a body to be in equilibrium,the net force $F$ acting on it must be zero. From the graph,the force $F$ is zero at points $P$ and $Q$.
For stable equilibrium,the potential energy $U$ must be at a minimum,which implies $\frac{d^2 U}{d x^2} > 0$.
Since $F = -\frac{d U}{d x}$,we have $\frac{d F}{d x} = -\frac{d^2 U}{d x^2}$.
Therefore,for stable equilibrium,$\frac{d F}{d x} < 0$,which means the slope of the $F-x$ graph must be negative.
At point $P$,the slope of the $F-x$ curve is positive.
At point $Q$,the slope of the $F-x$ curve is negative.
Thus,the body is in a stable equilibrium state at point $Q$.

(B) Assertion $A$ is true because a central force (like gravitational or electrostatic force) is a conservative force. By definition,the work done by a conservative force is independent of the path taken and depends only on the initial and final positions.
Reason $R$ is also true because non-conservative forces (like friction or air resistance) do not have an associated potential energy function.
However,the fact that some forces do not have potential energy is not the reason why central forces are path-independent. Path independence is a fundamental property of conservative forces themselves. Therefore,$R$ is not the correct explanation of $A$.

(C) Potential energy is defined only for conservative forces.
Conservative forces are those for which the work done in moving a particle between two points is independent of the path taken.
Examples of conservative forces include gravitational force,electrostatic (Coulomb's) force,and spring (restoring) force.
Frictional force is a non-conservative force because the work done against it depends on the path taken.
Therefore,potential energy cannot be defined for frictional force.

(A) $1$. Assertion $(A)$ is correct. By definition,a force is non-conservative if the work done by it depends on the path taken between two points. Conversely,for a conservative force,the work done is independent of the path and depends only on the initial and final positions.
$2$. Reason $(R)$ is also correct. $A$ central force is a force that acts along the line joining the centers of two objects. All conservative forces (like gravitational force and electrostatic force) are central forces.
$3$. However,the fact that conservative forces are central forces does not explain why a force is non-conservative based on path dependence. Therefore,$(R)$ is not the correct explanation of $(A)$.

(C) non-conservative force is a force for which the work done depends on the path taken between the initial and final positions.
Unlike conservative forces,the work done by a non-conservative force over a closed path is not zero.
Therefore,the statement that the work done by non-conservative forces depends on the path is correct.

(C) The work done by a conservative force is defined as $W_c = -\Delta U$,which is the negative change in potential energy. Thus,option $A$ is correct.
Total energy of a system is conserved only if the system is isolated (no external work or heat transfer). In general,the total energy of a system is not always conserved if external forces act on it. Thus,option $B$ is a general statement that is often considered incorrect in the context of non-isolated systems.
Non-conservative forces (like friction) dissipate energy. The work done by a non-conservative force in a closed path is not zero. Thus,option $C$ is definitely an incorrect statement.
In stable equilibrium,the potential energy $U$ is at a local minimum,meaning $dU/dx = 0$ and $d^2U/dx^2 > 0$. Thus,option $D$ is correct.
Comparing $B$ and $C$,$C$ is a fundamental property of non-conservative forces (they are path-dependent),making it explicitly false. Therefore,$C$ is the intended incorrect statement.

(B) For a conservative force,the relationship between force $F$ and potential energy $U$ is given by $F = -\frac{dU}{dx}$.
Given $U(x) = 5x^2 - 4x^3$.
To find the position where the force is zero,we set $F = 0$:
$0 = -\frac{d}{dx}(5x^2 - 4x^3)$
$0 = -(10x - 12x^2)$
$12x^2 - 10x = 0$
$2x(6x - 5) = 0$
This gives two solutions: $x = 0 \ m$ or $x = 5/6 \ m$.
Comparing this with the given options,the correct position is $5/6 \ m$.

(A) The relationship between conservative force $F$ and potential energy $U(x)$ is given by $F = -\frac{dU}{dx}$.
This means the force is equal to the negative of the slope of the $U(x)$ versus $x$ graph.
For the region $OA$ (from $x=0$ to $x=P$),the graph is a straight line with a positive constant slope. Let the slope be $k > 0$. Thus,$F = -k$,which is a negative constant.
For the region $AB$ (for $x > P$),the graph is a horizontal straight line. The slope of this line is $0$. Thus,$F = -(0) = 0$.
Comparing this with the given options,the graph that shows a negative constant force for $x < P$ and zero force for $x > P$ is represented by option $A$.

(A) conservative force is defined as a force where the total work done in moving a particle between two points is independent of the path taken.
It depends solely on the initial and final positions of the particle.
Mathematically,the work done by a conservative force around any closed path is zero.
Therefore,the correct option is $A$.

(D) Statement-$I$: Incorrect.
The work done by a force $\vec{F}$ is defined as $W = \int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$. The potential energy change $\Delta U$ is defined as $-\int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$. Thus,the given expression for work is incorrect.
Statement-$II$: Incorrect.
$A$ conservative force is defined as a force for which the work done is independent of the path taken between two points. Therefore,the work done does not change with the path followed.