Example 1: Calculate the gravitational force if the mass of the sun is \( 1.99 \times 10^{\wedge} 30 \mathrm{~kg} \) and earth is \( 5.97 \times 10^{\wedge} 24 \mathrm{~kg} \) separated by the distance \( 1.5 \times 10^{\wedge} 11 \mathrm{~m} \) ?(Gravitational constant \( \mathrm{G}=6.673 \times 10^{\wedge}-11 \mathrm{Nm} 2 / \) Kg 2 ) \( \quad \) Edited 19:23

Certainly! Let's calculate the gravitational force between the Sun and the Earth using Newton's **Universal Law of Gravitation**. Here's a step-by-step guide to solving the problem. ## **Given Data** - **Mass of the Sun (M₁):** \( 1.99 \times 10^{30} \) kg - **Mass of the Earth (M₂):** \( 5.97 \times 10^{24} \) kg - **Distance between Sun and Earth (r):** \( 1.5 \times 10^{11} \) meters - **Gravitational Constant (G):** \( 6.673 \times 10^{-11} \) N·m²/kg² ## **Formula** Newton's Universal Law of Gravitation is given by: \[ F = G \times \frac{M_1 \times M_2}{r^2} \] Where: - \( F \) is the gravitational force, - \( G \) is the gravitational constant, - \( M_1 \) and \( M_2 \) are the masses of the two objects, - \( r \) is the distance between the centers of the two masses. ## **Calculation Steps** 1. **Square the Distance (r²):** \[ r^2 = (1.5 \times 10^{11} \, \text{m})^2 = 2.25 \times 10^{22} \, \text{m}^2 \] 2. **Multiply the Masses (M₁ × M₂):** \[ M_1 \times M_2 = (1.99 \times 10^{30} \, \text{kg}) \times (5.97 \times 10^{24} \, \text{kg}) = 1.19 \times 10^{55} \, \text{kg}^2 \] 3. **Calculate the Gravitational Force (F):** \[ F = 6.673 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \times \frac{1.19 \times 10^{55} \, \text{kg}^2}{2.25 \times 10^{22} \, \text{m}^2} \] \[ F = 6.673 \times 10^{-11} \times 5.2889 \times 10^{32} \, \text{N} \] \[ F = 3.535 \times 10^{22} \, \text{N} \] ## **Final Answer** The gravitational force between the Sun and the Earth is approximately: \[ F \approx 3.54 \times 10^{22} \, \text{Newtons} \] ## **Understanding the Result** A force of \( 3.54 \times 10^{22} \, \text{N} \) is incredibly large, which is consistent with the massive sizes and masses of celestial bodies like the Sun and Earth. This force keeps the Earth in its orbital path around the Sun, balancing the centripetal force required for its circular motion.