Gravitational field and acceleration due to gravity

 

Gravitational Field

Acceleration Due to Gravity (g)

The force of gravity is the pull of attraction between the Earth and objects on its surface. This force causes objects to accelerate towards the Earth when they fall freely.

In a given region, such as a laboratory, the acceleration due to gravity (g) is considered constant. It is given by the formula:

$$g = \frac{G M}{R^2}$$

Where:

• G = Gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$)
• M = Mass of the Earth ($5.972 \times 10^{24} \, \text{kg}$)
• R = Radius of the Earth ($6.371 \times 10^6 \, \text{m}$)

The standard value of g near the Earth's surface is approximately:

$$g \approx 9.8 \, \text{m/s}^2$$

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Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation states that:

"Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers."

Mathematically,

$$F \propto \frac{m_1 m_2}{r^2}$$

By introducing the gravitational constant G, we get the equation:

$$F = G \frac{m_1 m_2}{r^2}$$

Where:

• F = Gravitational force (N)
• m₁, m₂ = Masses of the two objects (kg)
• r = Distance between their centers (m)
• G = Gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$)

This law explains the gravitational attraction between objects, including the force that keeps planets in orbit around the Sun.

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Worked Examples on Gravitational Field and Newton’s Law of Universal Gravitation

Example 1: Calculating Gravitational Force Between Two Objects

Question:
Two objects of masses 5 kg and 10 kg are placed 2 meters apart. Calculate the gravitational force between them. (Take G = $6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$).

Solution:
Using Newton’s Law of Universal Gravitation:

$$F = G \frac{m_1 m_2}{r^2}$$

Substituting the given values:

$$F = (6.674 \times 10^{-11}) \times \frac{(5 \times 10)}{(2)^2}$$ $$F = (6.674 \times 10^{-11}) \times \frac{50}{4}$$ $$F = (6.674 \times 10^{-11}) \times 12.5$$ $$F = 8.34 \times 10^{-10} \text{ N}$$

Final Answer:
The gravitational force between the two objects is $8.34 \times 10^{-10} \, N$.

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Example 2: Finding Acceleration Due to Gravity (g) on a Planet

Question:
A planet has a mass of $6.0 \times 10^{24} \, \text{kg}$ and a radius of $6.4 \times 10^6 \, \text{m}$. Find the acceleration due to gravity on its surface. (Take G = $6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}$).

Solution:
We use the formula:

$$g = \frac{G M}{R^2}$$

Substituting the values:

$$g = \frac{(6.674 \times 10^{-11}) \times (6.0 \times 10^{24})}{(6.4 \times 10^6)^2}$$ $$g = \frac{(4.0044 \times 10^{14})}{(4.096 \times 10^{13})}$$ $$g = 9.78 \text{ m/s}^2$$

Final Answer:
The acceleration due to gravity on the planet is 9.78 m/s².

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Example 3: Weight of an Object on Another Planet

Question:
An object has a mass of 50 kg. If the acceleration due to gravity on the Moon is 1.62 m/s², calculate its weight on the Moon.

Solution:
Weight is given by:

$$W = mg$$

Substituting the given values:

$$W = 50 \times 1.62$$ $$W = 81 \text{ N}$$

Final Answer:
The weight of the object on the Moon is 81 N.

JAMB Quiz: Gravitational Field & Acceleration Due to Gravity

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