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Question

Gravitational potential at a point due to a mass $M$ at a distance $r$ from it is given by:

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
 **Primary Concept**: Gravitational Potential
 **Related Topics**: Gravitation, Potential Energy
 **Prerequisites**: Understanding of potential energy and gravitational force.
## SOLUTION PATHWAY
**Given Information**:
  Mass of the object: $M$
  Distance from the mass: $r$
**Method & Steps**:
 1. Define Gravitational Potential:
  Reasoning: Gravitational potential at a point is defined as the work done per unit mass in bringing a small test mass from infinity to that point against the gravitational force.
  Formula/Rule: $V = -\frac{GM}{r}$
 2. Derivation (optional):
  Working: The gravitational force between two masses $m$ and $M$ separated by a distance $r$ is given by Newton's law of gravitation: $F = -\frac{GMm}{r^2}$. The work done in bringing a mass $m$ from infinity to a distance $r$ is given by the integral of the force over the distance: $W = \int_{\infty}^{r} F dr = \int_{\infty}^{r} -\frac{GMm}{r^2} dr = \frac{GMm}{r}$.  The potential is the work done per unit mass: $V = \frac{W}{m} = -\frac{GM}{r}$ 
  Key Point: The negative sign indicates that the gravitational potential is negative, meaning that work is done *by* the gravitational field when a mass moves from infinity to a closer point.
 3. Final Answer:
  Result: The gravitational potential at a point due to a mass $M$ at a distance $r$ is $-\frac{GM}{r}$ 
  Verification: The units are consistent with potential energy per unit mass ($J/kg$).
 ## OPTION ANALYSIS
A :x: This is the magnitude of the gravitational field strength, not the potential.
B ✓: This is the correct formula for gravitational potential.
C :x: This is the magnitude of the gravitational force.
D :x: This is incorrect; it mixes the potential and force expressions.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Confusing gravitational potential with gravitational field strength: Remember that potential is energy per unit mass, while field strength is force per unit mass.
  2.Forgetting the negative sign: The negative sign is crucial and indicates the attractive nature of the gravitational force.
 **Quick Tips**:
  1.Remember the formula: $V = -\frac{GM}{r}$
  2.Relate the potential to the work done against gravity.

ABDUL JILANI BASHA · Answer

$\frac{GM}{r}$

ABDUL JILANI BASHA · Answer

The correct Option is "$-\frac{GM}{r}$"

ABDUL JILANI BASHA · Answer

$\frac{GM}{r^2}$

ABDUL JILANI BASHA · Answer

$-\frac{GM}{r^2}$

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