RRB NTPC & Group D Physics: Master Gravitation – Complete Guide with Solved Problems & Tricks

Introduction to Gravitation for RRB Exams

Gravitation is one of the most fundamental topics in Physics, holding immense weightage in competitive exams conducted by the Railway Recruitment Board (RRB), including RRB NTPC, Group D, and Technician posts. Understanding the attractive force between masses, planetary motions, and the behavior of objects under gravity is crucial not only for clearing basic science sections but also for scoring high. This comprehensive guide breaks down the core concepts of Gravitation into digestible segments, packed with formulas, shortcuts, and meticulously solved questions tailored for railway aspirants.

Topic Weightage and Importance

In RRB exams, General Science is a scoring section where Physics contributes a significant share of questions. Under Physics, mechanics and gravitation form the bedrock. Candidates can expect 2 to 3 direct or application-based questions from Gravitation in both Computer Based Tests (CBT) Stage 1 and Stage 2. Questions range from direct formula applications like finding acceleration due to gravity to conceptual queries regarding satellite motion, weightlessness, and mass versus weight. Mastering this topic ensures you secure those vital marks.

Key Concepts and Formulas

To tackle numerical and conceptual problems effortlessly, you must memorize and understand the core formulas associated with gravitation:

• Newton's Law of Gravitation: Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
  Formula: $F = \frac{G m_1 m_2}{r^2}$
  where $G$ is the Universal Gravitational Constant ($6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$).
• Acceleration due to Gravity ($g$):
  Formula: $g = \frac{G M}{R^2}$
  where $M$ is the mass of the Earth and $R$ is the radius of the Earth. The standard value is $9.8 \text{ m/s}^2$.
• Variation of $g$ with Altitude ($h$):
  Formula: $g_h = g \left(1 - \frac{2h}{R}\right)$ (for small heights where $h \ll R$).
• Variation of $g$ with Depth ($d$):
  Formula: $g_d = g \left(1 - \frac{d}{R}\right)$.
• Kepler's Third Law of Planetary Motion: The square of the time period of revolution of a planet is directly proportional to the cube of the semi-major axis of its elliptical orbit.
  Formula: $T^2 \propto r^3$.
• Escape Velocity ($v_e$): The minimum velocity required by an object to escape Earth's gravitational pull.
  Formula: $v_e = \sqrt{2gR} \approx 11.2 \text{ km/s}$.

Solved Examples (Step-by-Step)

Example 1: Newton's Law of Gravitation

Problem: Two bodies of masses $10\text{ kg}$ and $20\text{ kg}$ are separated by a distance of $2\text{ m}$. Calculate the gravitational force between them. (Take $G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$)

Solution:
Step 1: Identify the given values.
$m_1 = 10\text{ kg}$, $m_2 = 20\text{ kg}$, $r = 2\text{ m}$, $G = 6.67 \times 10^{-11}\text{ N m}^2/\text{kg}^2$.
Step 2: Apply Newton's Law of Gravitation formula.
$F = \frac{G m_1 m_2}{r^2}$
Step 3: Substitute the values.
$F = \frac{6.67 \times 10^{-11} \times 10 \times 20}{2^2}$
$F = \frac{6.67 \times 10^{-11} \times 200}{4} = 6.67 \times 10^{-11} \times 50$
$F = 3.335 \times 10^{-9}\text{ N}$.
Answer: The gravitational force is $3.335 \times 10^{-9}\text{ N}$.

Example 2: Variation of 'g'

Problem: What will be the value of acceleration due to gravity at a height equal to the radius of the Earth ($R$) from the Earth's surface?

Solution:
Step 1: Use the exact formula for variation of $g$ with height $h$: $g_h = \frac{G M}{(R + h)^2}$.
Step 2: Substitute $h = R$.
$g_h = \frac{G M}{(R + R)^2} = \frac{G M}{(2R)^2} = \frac{G M}{4R^2}$
Step 3: Since $g = \frac{G M}{R^2}$, we get:
$g_h = \frac{g}{4}$.
Answer: The value of $g$ becomes one-fourth of its value at the Earth's surface.

Example 3: Escape Velocity Calculation

Problem: If the radius of a hypothetical planet is twice that of Earth and its mass is four times that of Earth, find the escape velocity on that planet if Earth's escape velocity is $11.2\text{ km/s}$.

Solution:
Step 1: Write the formula for escape velocity: $v_e = \sqrt{\frac{2GM}{R}}$.
Step 2: Set up the ratio for the planet ($p$) and Earth ($e$).
$\frac{v_{ep}}{v_{ee}} = \sqrt{\frac{M_p}{M_e} \times \frac{R_e}{R_p}}$
Step 3: Substitute given relations: $M_p = 4M_e$ and $R_p = 2R_e$.
$\frac{v_{ep}}{v_{ee}} = \sqrt{4 \times \frac{1}{2}} = \sqrt{2}$
Step 4: Calculate $v_{ep}$.
$v_{ep} = \sqrt{2} \times 11.2\text{ km/s} \approx 1.414 \times 11.2 \approx 15.84\text{ km/s}$.
Answer: The escape velocity on the planet is $15.84\text{ km/s}$.

Common Mistakes to Avoid

• Confusing Mass and Weight: Mass is constant everywhere, while weight ($W = mg$) changes depending on the value of $g$ at that location.
• Ignoring Unit Conversions: Always convert distances into meters (m) and masses into kilograms (kg) before applying the gravitational formula.
• Misapplying approximations for height: The formula $g_h = g(1 - \frac{2h}{R})$ is only valid when $h$ is very small compared to $R$. For higher altitudes, use the full fractional formula.
• Forgetting that the value of $g$ is zero at the center of the Earth ($d = R$).

Practice Questions with Solutions

Q1: If the distance between two masses is doubled, the gravitational force between them becomes:

Q2: Where is the value of acceleration due to gravity ($g$) maximum on Earth's surface?

Q3: What is the time period of a geostationary satellite?

Q4: If the Earth suddenly stops rotating, what will happen to the weight of a body at the equator?

Q5: The orbital velocity of a satellite close to the Earth's surface depends on:

Solutions to Practice Questions:
Sol 1: Since $F \propto \frac{1}{r^2}$, doubling the distance ($2r$) makes the force $\frac{1}{2^2} = \frac{1}{4}$th of the original value.
Sol 2: At the poles, because the Earth is flattened at the poles (smaller radius) and centrifugal force due to rotation is zero.
Sol 3: 24 hours, matching the rotational period of the Earth.
Sol 4: The weight will increase because the centrifugal force acting outward at the equator will become zero.
Sol 5: It depends on the mass and radius of the Earth, given by $v_o = \sqrt{gR}$, and is independent of the mass of the satellite.

Frequently Asked Questions (FAQs)

• Q: Is the gravitational constant ($G$) the same everywhere in the universe?
  A: Yes, $G$ is a universal constant and does not depend on the medium, temperature, or location.
• Q: Why do astronauts float in space?
  A: Astronauts float because they are in a state of free fall along with their spacecraft, experiencing apparent weightlessness.
• Q: Does a heavy body fall faster than a lighter body in a vacuum?
  A: No, acceleration due to gravity is independent of mass. In a vacuum, all bodies fall at the same rate.

Conclusion and Final Tips

Gravitation is a high-yield topic that bridges conceptual understanding with simple numerical calculation. To ace questions in RRB NTPC and Group D exams, focus on memorizing standard formulas, understanding directional variations of $g$, and practicing direct ratio-based numericals. Keep revising these core principles regularly, and you will easily clear the General Science section with flying colors. Good luck with your preparation!