Gravitational Potential — Revision Notes

⚡ 30-Second Revision

Definition: — Work done by external agent to bring unit mass from $\infty$ to point (J/kg).\n- Sign: Always negative (attractive force, bound system).\n- Scalar: Yes, add algebraically.\n- Point Mass: $V = -\frac{GM}{r}$\n- **Spherical Shell (Mass $M$, Radius $R$):**\n - Outside ($r \ge R$): $V = -\frac{GM}{r}$\n - Inside ($r < R$): $V = -\frac{GM}{R}$ (constant)\n- **Solid Sphere (Mass $M$, Radius $R$):**\n - Outside ($r \ge R$): $V = -\frac{GM}{r}$\n - Inside ($r < R$): $V = -\frac{GM}{2R^3}(3R^2 - r^2)$\n - Center ($r=0$): $V_{\text{center}} = -\frac{3GM}{2R}$\n- Relation to Potential Energy: $U = mV$\n- Relation to Field Intensity: $\vec{E} = -\nabla V$ (Field points towards decreasing potential)\n- Escape Velocity: $v_{\text{esc}} = \sqrt{-2V}$

2-Minute Revision

Gravitational potential ($V$) is a crucial scalar quantity in gravitation, representing the work done per unit mass (J/kg) to bring a test mass from infinity to a specific point without acceleration. It's always negative, signifying the attractive nature of gravity and a bound system.

The reference point for zero potential is conventionally taken at infinity. For a point mass $M$, the potential at distance $r$ is $V = -GM/r$. This formula is foundational. For a spherical shell of mass $M$ and radius $R$, the potential outside is $-GM/r$, but crucially, it's constant inside and equal to $-GM/R$.

For a solid sphere, it's $-GM/r$ outside, but inside, it varies parabolically, reaching its most negative value (minimum) at the center, $V_{\text{center}} = -3GM/(2R)$. Gravitational potential energy ($U$) for a mass $m$ at a point with potential $V$ is simply $U = mV$.

The gravitational field intensity $\vec{E}$ is related to potential by $\vec{E} = -\nabla V$, meaning the field points in the direction of decreasing potential. Escape velocity from a point can be directly calculated from its potential using $v_{\text{esc}} = \sqrt{-2V}$.

Remember to use correct signs and units in all calculations.

5-Minute Revision

Gravitational potential, $V$, is a scalar measure of the gravitational field's influence at a point, defined as the work done by an external agent to bring a unit test mass from infinity to that point without acceleration.

Its unit is J/kg. By convention, potential at infinity is zero. Since gravity is attractive, work is done by the field, making the potential negative. A more negative potential implies a stronger binding.

This scalar nature simplifies calculations for multiple masses, as potentials can be algebraically summed.\n\nKey formulas to remember:\n1. **Point Mass $M$:** $V = -\frac{GM}{r}$. This is the basic building block.

\n2. **Spherical Shell (Mass $M$, Radius $R$):**\n * Outside ($r \ge R$): $V = -\frac{GM}{r}$\n * Inside ($r < R$): $V = -\frac{GM}{R}$ (constant, equal to surface potential)\n3. **Solid Sphere (Mass $M$, Radius $R$):**\n * Outside ($r \ge R$): $V = -\frac{GM}{r}$\n * Inside ($r < R$): $V = -\frac{GM}{2R^3}(3R^2 - r^2)$.

At the center ($r=0$), $V_{\text{center}} = -\frac{3GM}{2R}$.\n\nRelationship with Potential Energy: The gravitational potential energy $U$ of a mass $m$ at a point with potential $V$ is $U = mV$.

The work done by an external agent to move a mass $m$ from point A to point B is $W_{\text{ext}} = \Delta U = m(V_B - V_A)$.\n\nRelationship with Field Intensity: The gravitational field intensity $\vec{E}$ is the negative gradient of the potential: $\vec{E} = -\nabla V$.

In one dimension, $E = -dV/dr$. This means the gravitational field always points in the direction of decreasing potential.\n\nEscape Velocity: The minimum velocity required for an object to escape a gravitational field from a point with potential $V$ is $v_{\text{esc}} = \sqrt{-2V}$.

\n\nExample: Consider a solid sphere of mass $M=6 \times 10^{24}\,\text{kg}$ and radius $R=6.4 \times 10^6\,\text{m}$ (Earth). Calculate potential at its center.\n$V_{\text{center}} = -\frac{3GM}{2R} = -\frac{3 \times (6.

67 \times 10^{-11}\,\text{Nm}^2/\text{kg}^2) \times (6 \times 10^{24}\,\text{kg})}{2 \times (6.4 \times 10^6\,\text{m})} \approx -6.25 \times 10^7\,\text{J/kg}$.\n\nCommon pitfalls include confusing potential with potential energy, making sign errors, or misapplying formulas for different mass distributions.

Always remember the scalar nature for superposition and the negative gradient for field-potential relation.

Prelims Revision Notes

Gravitational Potential (V)\n* Definition: Work done by an external agent to bring a unit test mass from infinity to a point in a gravitational field without acceleration.\n* Nature: Scalar quantity. Add algebraically for multiple masses.\n* Unit: Joules per kilogram (J/kg).\n* Sign Convention: Always negative (assuming $V=0$ at infinity). Negative sign indicates attractive force and a bound system. More negative means stronger binding.\n\n### Formulas for Gravitational Potential:\n1. Due to a Point Mass $M$:\n $V = -\frac{GM}{r}$\n where $r$ is the distance from the point mass.\n\n2. Due to a Spherical Shell (Mass $M$, Radius $R$):\n * Outside the shell ($r \ge R$): $V = -\frac{GM}{r}$\n * Inside the shell ($r < R$): $V = -\frac{GM}{R}$ (constant, equal to potential at the surface)\n\n3. Due to a Solid Sphere (Mass $M$, Radius $R$, uniform density):\n * Outside the sphere ($r \ge R$): $V = -\frac{GM}{r}$\n * Inside the sphere ($r < R$): $V = -\frac{GM}{2R^3}(3R^2 - r^2)$\n * At the center ($r=0$): $V_{\text{center}} = -\frac{3GM}{2R}$ (most negative potential)\n * At the surface ($r=R$): $V_{\text{surface}} = -\frac{GM}{R}$\n\n### Key Relationships:\n* Gravitational Potential Energy ($U$): For a mass $m$ at a point where potential is $V$, $U = mV$. (Unit: Joules, J)\n* Work Done ($W$): Work done by an external agent to move mass $m$ from point A (potential $V_A$) to point B (potential $V_B$) is $W_{\text{ext}} = \Delta U = U_B - U_A = m(V_B - V_A)$.\n* Gravitational Field Intensity ($\vec{E}$): $\vec{E} = -\nabla V$. In one dimension, $E = -\frac{dV}{dr}$. This implies that the gravitational field points in the direction of decreasing potential.\n* Escape Velocity ($v_{\text{esc}}$): The minimum speed required to escape a gravitational field from a point with potential $V$ is $v_{\text{esc}} = \sqrt{-2V}$. (Note: $V$ is negative, so $-2V$ is positive, ensuring a real value for $v_{\text{esc}}$).\n\n### Common Traps & Tips:\n* Sign Errors: Always be careful with the negative sign in potential formulas and when calculating changes in potential energy.\n* Confusion: Do not confuse gravitational potential ($V$, J/kg) with gravitational potential energy ($U$, J) or gravitational field intensity ($\vec{E}$, N/kg or m/s$^2$).\n* Graphs: Understand the shape of $V$ vs $r$ graphs for different mass distributions. For a solid sphere, $V$ is parabolic inside and hyperbolic outside.\n* Superposition: For multiple point masses, simply add the scalar potentials algebraically.

Vyyuha Quick Recall

Very Negative Gravity Makes Radius Small. (V = -GM/r) - Helps recall the point mass formula and negative sign. \n\nSolid Sphere Center 3/2 Surface. (V_center = 3/2 * V_surface) - Reminds the relation between center and surface potential for a solid sphere.