Introduction to Gravitation and Gravity for RRB Exams
Gravitation is one of the most fundamental forces in the universe and a cornerstone subject in General Science for competitive exams conducted by the Railway Recruitment Board (RRB). Whether you are appearing for RRB NTPC, RRB Group D, RRB Technician Grade I, or Grade III, conceptual understanding and numerical problem-solving related to Gravitation and Gravity are crucial. Understanding how planets move, why objects fall toward the Earth, how mass differs from weight, and how acceleration due to gravity varies across different locations can help you secure high scores in the Physics section.
Topic Weightage and Importance
In RRB examinations, General Science accounts for a significant portion of the total score. Specifically, Physics questions carry substantial weightage:
- RRB Group D: General Science consists of 25 questions, out of which 2 to 4 direct conceptual and numerical questions are asked on Gravitation, Kepler's Laws, Mass vs. Weight, and variation of acceleration due to gravity (\(g\)).
- RRB NTPC (CBT-1 & CBT-2): Expect 1 to 3 questions related to gravitational formulas, escape velocity, satellite motion, and planetary laws.
- RRB Technician (Grade I & Grade III): Given the technical nature of these exams, basic numerical calculations involving gravitational force, acceleration due to gravity at heights/depths, and weight changes in elevators are standard test items.
Key Concepts and Formulas
1. Newton's Universal Law of Gravitation
Every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Formula: \(F = G \frac{m_1 m_2}{r^2}\)
- \(F\) = Gravitational force between two objects (in Newtons, N)
- \(m_1, m_2\) = Masses of the two objects (in kg)
- \(r\) = Distance between the centers of the two masses (in meters, m)
- \(G\) = Universal Gravitational Constant = \(6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2\)
Important Note: The value of \(G\) was experimentally measured by Henry Cavendish and remains constant throughout the universe.
2. Acceleration Due to Gravity (g)
Gravity is the attractive force exerted by a large celestial body (like Earth) on objects near its surface. The acceleration produced in a body due to this gravitational pull is called acceleration due to gravity (\(g\)).
Formula: \(g = \frac{G M}{R^2}\)
- \(M\) = Mass of Earth (\(\approx 6 \times 10^{24} \text{ kg}\))
- \(R\) = Radius of Earth (\(\approx 6.4 \times 10^6 \text{ m}\))
- Standard value of \(g\) on Earth's surface = \(9.8 \text{ m/s}^2\) (often approximated as \(10 \text{ m/s}^2\) in exam problems).
3. Factors Affecting Acceleration Due to Gravity (g)
| Condition / Location | Effect on 'g' | Formula / Reason |
|---|---|---|
| Altitude (Height 'h' above surface) | Decreases with height | \(g' = g \left(1 - \frac{2h}{R}\right)\) for \(h \ll R\) |
| Depth ('d' below surface) | Decreases with depth | \(g' = g \left(1 - \frac{d}{R}\right)\) |
| At Earth's Center | Becomes Zero | \(d = R \implies g' = 0\) |
| Earth's Shape (Poles vs Equator) | Maximum at Poles, Minimum at Equator | Earth is flattened at poles ( smaller \(R\) ) and bulged at equator ( larger \(R\) ) |
| Earth's Rotation | Decreases due to centrifugal action | \(g' = g - \omega^2 R \cos^2 \lambda\) (Effect is max at equator, zero at poles) |
4. Difference Between Mass and Weight
| Property | Mass (m) | Weight (W) |
|---|---|---|
| Definition | Amount of matter contained in a body | Gravitational force exerted by Earth on a body |
| Type of Quantity | Scalar Quantity | Vector Quantity (directed towards center of Earth) |
| SI Unit | Kilogram (kg) | Newton (N) or kg-wt |
| Variability | Constant everywhere in universe | Varies depending on the value of \(g\) (\(W = m \times g\)) |
| Value at Earth's Center | Non-zero (Unchanged) | Zero (since \(g = 0\)) |
5. Kepler's Laws of Planetary Motion
- First Law (Law of Orbits): All planets move in elliptical orbits with the Sun located at one of the two foci.
- Second Law (Law of Areas): A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time (Areal velocity is constant; Angular momentum is conserved).
- Third Law (Law of Periods): The square of the orbital period (\(T\)) of a planet is directly proportional to the cube of the semi-major axis (\(a\)) of its orbit: \(T^2 \propto a^3\).
6. Escape Velocity and Orbital Velocity
- Orbital Velocity (\(v_o\)): Speed required for a satellite to revolve around Earth in a close orbit: \(v_o = \sqrt{g R} \approx 7.92 \text{ km/s}\).
- Escape Velocity (\(v_e\)): Minimum speed required for an object to escape Earth's gravitational field permanently: \(v_e = \sqrt{2 g R} = \sqrt{2} v_o \approx 11.2 \text{ km/s}\).
Solved Examples (Step-by-Step)
Example 1:
Question: An object has a mass of 60 kg on Earth. What will be its mass and weight on the surface of the Moon? (Take \(g_{\text{Earth}} = 10 \text{ m/s}^2\) and gravity on Moon = \(\frac{1}{6}\text{th}\) of Earth's gravity).
Solution:
- Mass on Earth: \(m = 60 \text{ kg}\)
- Since mass is constant everywhere, Mass on Moon = 60 kg.
- Acceleration due to gravity on Moon (\(g_{\text{Moon}}\)): \(g_{\text{Moon}} = \frac{10}{6} = 1.67 \text{ m/s}^2\)
- Weight on Moon (\(W_{\text{Moon}}\)): \(W_{\text{Moon}} = m \times g_{\text{Moon}} = 60 \times \frac{10}{6} = 100 \text{ N}\)
Answer: Mass = 60 kg, Weight = 100 N.
Example 2:
Question: If the distance between two masses is doubled, how does the gravitational force between them change?
Solution:
- Initial Gravitational Force: \(F_1 = G \frac{m_1 m_2}{r^2}\)
- New distance \(r' = 2r\)
- New Force: \(F_2 = G \frac{m_1 m_2}{(2r)^2} = G \frac{m_1 m_2}{4 r^2} = \frac{F_1}{4}\)
Answer: The gravitational force becomes one-fourth (\(\frac{1}{4}\text{th}\)) of its original value.
Example 3:
Question: Calculate the height above Earth's surface where acceleration due to gravity becomes half of its value at the surface. (Radius of Earth = \(R\))
Solution:
- Formula for variation with height: \(g' = g \left(\frac{R}{R+h}\right)^2\)
- Given \(g' = \frac{g}{2}\)
- \(\frac{g}{2} = g \left(\frac{R}{R+h}\right)^2 \implies \frac{1}{2} = \left(\frac{R}{R+h}\right)^2\)
- Taking square root on both sides: \(\frac{1}{\sqrt{2}} = \frac{R}{R+h}\)
- \(R + h = R \sqrt{2} \implies h = R(\sqrt{2} - 1) \approx 0.414 R\)
Answer: At a height of \(h = (\sqrt{2} - 1) R \approx 0.414 R\).
Common Mistakes to Avoid
- Confusing Mass and Weight: Remembering that mass stays unchanged (60 kg remains 60 kg on Earth, Moon, or outer space), whereas weight changes depending on local gravitational acceleration \(g\).
- Mixing units of G and g: Universal Gravitational Constant \(G\) (\(\text{N}\cdot\text{m}^2/\text{kg}^2\)) is a universal constant, while \(g\) (\(\text{m/s}^2\)) is local acceleration due to gravity.
- Ignoring inverse square relationship: Doubling distance reduces force by a factor of 4, tripling distance reduces force by a factor of 9.
- Incorrect value of escape velocity relationship: Escape velocity is \(\sqrt{2}\) times orbital velocity (\(v_e = \sqrt{2} v_o\)), not double or half.
Practice Questions with Solutions
Question 1:
Where is the value of acceleration due to gravity (\(g\)) maximum on Earth's surface?
- A) At the Equator
- B) At the Poles
- C) At the Center of Earth
- D) Equal everywhere
Question 2:
What is the ratio of escape velocity to orbital velocity for a satellite close to Earth's surface?
- A) 1 : 1
- B) 1 : \(\sqrt{2}\)
- C) \(\sqrt{2}\) : 1
- D) 2 : 1
Question 3:
A body weighs 600 N on the surface of Earth. How much will it weigh at the center of Earth?
- A) 600 N
- B) 300 N
- C) 100 N
- D) Zero
Question 4:
If the radius of Earth shrinks by 1% while its mass remains constant, how will acceleration due to gravity (\(g\)) change?
- A) Increases by 1%
- B) Increases by 2%
- C) Decreases by 2%
- D) Decreases by 1%
Question 5:
Which law states that the areal velocity of a planet revolving around the Sun remains constant?
- A) Kepler's First Law
- B) Kepler's Second Law
- C) Kepler's Third Law
- D) Newton's Third Law
Solutions:
1. Answer: B) At the Poles
Explanation: The radius of Earth at the poles is smaller than at the equator due to Earth's oblate spheroid shape. Since \(g = \frac{GM}{R^2}\), a smaller radius leads to a higher value of \(g\) at the poles.
2. Answer: C) \(\sqrt{2}\) : 1
Explanation: \(v_e = \sqrt{2gR}\) and \(v_o = \sqrt{gR}\). Therefore, \(\frac{v_e}{v_o} = \frac{\sqrt{2gR}}{\sqrt{gR}} = \sqrt{2}\), giving a ratio of \(\sqrt{2} : 1\).
3. Answer: D) Zero
Explanation: At the center of Earth, acceleration due to gravity \(g = 0\). Therefore, weight \(W = m \times 0 = 0 \text{ N}\).
4. Answer: B) Increases by 2%
Explanation: \(g = \frac{GM}{R^2}\). For small percentage changes, \(\frac{\Delta g}{g} = -2 \frac{\Delta R}{R}\). If \(R\) decreases by 1%, \(g\) increases by \(2 \times 1\% = 2\%\).
5. Answer: B) Kepler's Second Law
Explanation: Kepler's Second Law (Law of Areas) states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, meaning areal velocity is constant.
Frequently Asked Questions (FAQs)
1. What is the value of Universal Gravitational Constant (G)?
The value of \(G\) is \(6.674 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2\). It is constant throughout the universe and does not depend on medium or temperature.
2. Why does a feather fall slower than a heavy stone in air?
In the presence of air, air resistance opposes motion. Due to its higher surface-area-to-mass ratio, the feather experiences greater relative drag. In a vacuum, both fall at the exact same rate because acceleration due to gravity is independent of the mass of the falling object.
3. What happens to weight in a freely falling elevator?
In a freely falling elevator, effective acceleration is \(g' = g - a = g - g = 0\). Therefore, apparent weight \(W = m g' = 0\), causing a state of weightlessness.
Conclusion and Final Tips
Gravitation is a high-yield concept for RRB NTPC, Group D, and Technician examinations. Focus on mastering core definitions, variations of \(g\) with height, depth, and latitude, key differences between mass and weight, and Kepler's laws. Make sure to practice standard numerical problems involving ratio changes when distance or radius is altered. Regular revision and solving past RRB Physics questions will help you secure full marks in this section!
