Gravitation | Class 11 Physics Notes

7.3 Universal law of gravitation

Newton's universal law of gravitation unifies terrestrial gravity and celestial mechanics. Inspired by the observation of an apple falling and the Moon's orbital motion, Newton proposed that every mass attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Newton reasoned that the Moon's centripetal acceleration (a_m) due to Earth's gravity is given by a_m = V² / R_m = 4π² R_m / T², where V is the Moon's orbital speed, R_m its orbital radius, and T its orbital period (~27.3 days). This acceleration is much smaller than the acceleration due to gravity (g) on Earth's surface, indicating that gravitational force decreases with distance.

Assuming inverse square dependence, g / a_m = (R_m / R_E)² ≈ 3600, consistent with measured values of g and a_m. This led Newton to state the Universal Law of Gravitation:

"Every body in the universe attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them."

Mathematically, the magnitude of the gravitational force F between two point masses m_1 and m_2 separated by distance r is:

F = G (m_1 m_2) / r²

where G is the universal gravitational constant. The force vector points along the line joining the masses and is attractive.

For extended bodies, the total gravitational force is the vector sum of forces due to all constituent point masses. Two important results simplify calculations:

1. A hollow spherical shell attracts an external point mass as if all its mass were concentrated at its center.

2. The gravitational force inside a hollow spherical shell is zero.

Example 7.2 applies the principle of superposition to calculate the net gravitational force on a mass placed at the centroid of an equilateral triangle formed by three equal masses, showing the resultant force is zero due to symmetry.

📊 Diagram: Fig. 7.3: Vector diagram showing gravitational force on mass m_1 due to m_2 along vector r = r_2 - r_1. Fig. 7.4: Multiple masses exerting gravitational forces on a point mass, illustrating vector addition. Fig. 7.5: Three equal masses at vertices of an equilateral triangle with a mass 2m at the centroid.

🧪 Activity: No specific activity, but includes detailed vector force addition example.

🔗 Connection: Leads to Section 7.4 where the gravitational constant G is experimentally determined.