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Imagine you are holding a ball in your hand. You feel a force pulling it down towards the ground. This force is called weight, and it is caused by the gravitational force exerted by the Earth on the ball. But what exactly is gravitational force? How is it related to weight? And how does weight change if you go to the Moon or climb a mountain?
In this section, we will explore these questions step-by-step, starting from the basic idea of gravitational force as an attraction between masses, then defining weight as the force of gravity acting on an object, and finally understanding the relationship between mass and weight. We will use simple examples, clear diagrams, and formulas to build a solid understanding that will help you solve related problems confidently.
Gravitational force is a natural force of attraction that exists between any two objects that have mass. This means every object in the universe pulls on every other object with a force that depends on their masses and the distance between them.
Sir Isaac Newton formulated this idea in his Law of Universal Gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
This formula tells us three important things:
Let's visualize this with a diagram:
Here, the two spheres represent masses \( M_1 \) and \( M_2 \), separated by distance \( r \). The arrows labeled \( F \) show the gravitational forces acting on each mass, directed towards each other.
While gravitational force exists between any two masses, when we talk about the weight of an object, we mean the gravitational force exerted by the Earth on that object. Weight is the force with which the Earth pulls the object towards its center.
Weight depends on two things:
The formula for weight is:
On Earth, \( g \) is approximately \( 9.8 \, \mathrm{m/s^2} \), but it varies slightly depending on location and altitude.
Let's see how weight acts on an object with a diagram:
In this diagram, the green block represents an object resting on a surface. The downward arrow shows the weight \( W = mg \), and the upward red arrow shows the normal force exerted by the surface, balancing the weight when the object is at rest.
It is important to understand that mass and weight are related but different physical quantities:
For example, a person with a mass of 70 kg has the same mass on Earth, Moon, or anywhere else. However, their weight changes because the Moon's gravity is weaker than Earth's.
| Location | Acceleration due to Gravity \( g \) (m/s²) | Weight of 70 kg person (N) |
|---|---|---|
| Earth | 9.8 | \( 70 \times 9.8 = 686 \, \mathrm{N} \) |
| Moon | 1.62 | \( 70 \times 1.62 = 113.4 \, \mathrm{N} \) |
This shows that weight depends on location, but mass does not.
Step 1: Write down the known values:
Step 2: Use Newton's law of gravitation:
\[ F = G \frac{m_1 m_2}{r^2} = 6.674 \times 10^{-11} \times \frac{5 \times 5}{2^2} \]
Step 3: Calculate the denominator:
\( 2^2 = 4 \)
Step 4: Calculate numerator:
\( 5 \times 5 = 25 \)
Step 5: Substitute values:
\[ F = 6.674 \times 10^{-11} \times \frac{25}{4} = 6.674 \times 10^{-11} \times 6.25 = 4.17125 \times 10^{-10} \, \mathrm{N} \]
Answer: The gravitational force between the two masses is approximately \( 4.17 \times 10^{-10} \, \mathrm{N} \).
Step 1: Write down known values:
Step 2: Calculate weight on Earth:
\[ W_{\text{Earth}} = mg = 10 \times 9.8 = 98 \, \mathrm{N} \]
Step 3: Calculate weight on Moon:
\[ W_{\text{Moon}} = mg = 10 \times 1.62 = 16.2 \, \mathrm{N} \]
Answer: The object weighs 98 N on Earth and 16.2 N on the Moon.
Step 1: Write down known values:
Step 2: Use the formula \( W = mg \) to find mass:
\[ m = \frac{W}{g} = \frac{196}{9.8} = 20 \, \mathrm{kg} \]
Answer: The mass of the object is 20 kg.
Step 1: Calculate acceleration due to gravity at height \( h = 1000 \, \mathrm{m} \) using:
\[ g_h = g \left(1 - \frac{h}{R}\right) \]
Step 2: Substitute values:
\[ g_h = 9.8 \times \left(1 - \frac{1000}{6.37 \times 10^6}\right) = 9.8 \times \left(1 - 1.57 \times 10^{-4}\right) \]
\[ g_h \approx 9.8 \times 0.999843 = 9.7986 \, \mathrm{m/s^2} \]
Step 3: Calculate weight at height:
\[ W_h = m g_h = 50 \times 9.7986 = 489.93 \, \mathrm{N} \]
Step 4: Calculate weight at surface:
\[ W = 50 \times 9.8 = 490 \, \mathrm{N} \]
Answer: The weight decreases slightly from 490 N to approximately 489.93 N at 1000 m height.
Step 1: Recall that acceleration due to gravity on a planet is:
\[ g = G \frac{M}{R^2} \]
where \( M \) is planet's mass and \( R \) is radius.
Step 2: Let Earth's gravity be \( g_E = 9.8 \, \mathrm{m/s^2} \).
Step 3: Since the new planet has mass \( M_p = 2 M_E \) and radius \( R_p = R_E \), its gravity is:
\[ g_p = G \frac{2 M_E}{R_E^2} = 2 \times g_E = 2 \times 9.8 = 19.6 \, \mathrm{m/s^2} \]
Step 4: Calculate weight on Earth:
\[ W_E = m g_E = 15 \times 9.8 = 147 \, \mathrm{N} \]
Step 5: Calculate weight on new planet:
\[ W_p = m g_p = 15 \times 19.6 = 294 \, \mathrm{N} \]
Answer: The object weighs 147 N on Earth and 294 N on the new planet, twice as much.
When to use: When solving problems involving weight on different celestial bodies.
When to use: To estimate weight changes with altitude without complex calculations.
When to use: During all calculations involving gravitational force and weight.
When to use: In time-limited competitive exams for quick recall.
When to use: To avoid unit-related mistakes in competitive exam problems.
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