Gravitation – D.Pharm Admission (Physics)
Target audience: Students preparing for IIT‑JEE / other competitive exams.
Goal: Build a deep, derivation‑based understanding of gravitation while keeping an eye on exam‑type problems.
1. Conceptual Overview
| Concept | Key Idea |
|---|---|
| Gravitation | A fundamental force that attracts all masses. |
| Newton’s Law of Universal Gravitation | Every pair of point masses attracts each other with a force proportional to the product of their masses and inversely proportional to the square of their separation. |
| Gravitational Field | Field quantity ( \mathbf{g}(\mathbf{r}) ) produced by a mass distribution, analogous to electric field. |
| Gravitational Potential | Scalar field ( V(\mathbf{r}) ) such that ( \mathbf{g} = -\nabla V ). |
| Gravitational Potential Energy (GPE) | Energy of a system of masses in a gravitational field. |
| Kepler’s Laws | Derived from Newtonian gravitation; describe planetary motion. |
| Vis‑Viva Equation | Relates orbital speed, radius, and specific mechanical energy: ( v^2 = GM!\left(\frac{2}{r} - \frac{1}{a}\right) ). |
2. Newton’s Law of Gravitation
2.1 Statement
[ \boxed{ \mathbf{F}{12} = G\frac{m_1 m_2}{r^2},\hat{\mathbf{r}}{12} } ]
- (G = 6.674\times10^{-11}, \mathrm{m^3,kg^{-1},s^{-2}})
- (r) is the separation between the two masses
- (\hat{\mathbf{r}}_{12}) is the unit vector from (m_1) to (m_2)
- The force on (m_2) is the negative of the force on (m_1) (Newton’s 3rd law).
2.2 Gravitational Field
For a point mass (M) at the origin:
[ \boxed{ \mathbf{g}(\mathbf{r}) = -,\frac{GM}{r^2},\hat{\mathbf{r}} } ]
- Units: ( \mathrm{m,s^{-2}} )
- Direction: Toward the mass (hence the minus sign).
3. Gravitational Potential
3.1 Definition
[ V(\mathbf{r}) = -\int_{\infty}^{\mathbf{r}} \mathbf{g}!\cdot d\mathbf{l} ]
For a point mass:
[ \boxed{ V(r) = -\frac{GM}{r} }, \qquad V(\infty)=0 ]
- Units: ( \mathrm{J,kg^{-1}} ) (equivalent to ( \mathrm{m^2,s^{-2}} )).
- Sign: Negative because work is done against gravity to bring a unit mass from infinity to (r).
3.2 Gravitational Potential Energy
For two point masses:
[ U_{12} = m_1 V_2 = -\frac{G m_1 m_2}{r} ]
The total GPE of a system is the sum over all pairs.
4. Derivations & Key Results
4.1 Shell Theorem (Key Tool)
- Outside a spherical shell of mass (M): field is identical to that of a point mass (M) at the centre.
- **Inside a spherical