Mechanics: Motion, Newton's Laws, Work, Energy, Power and Gravitation
Mechanics is the oldest and most fundamental branch of physics. It deals with the behaviour of physical bodies when they are subjected to forces or displacements, and with the subsequent effects of those bodies on their environment. Everything from a falling mango in a Guwahati garden to the orbital motion of the Moon around the Earth, and from the recoil of a rifle to the thrust of a rocket, is governed by the principles of mechanics. For students preparing for competitive examinations in Assam, such as the Assam Public Service Commission (APSC), the Assam Direct Recruitment Examination (ADRE), SSC, Railways, NEET and JEE, mechanics forms the backbone of the physics section. A clear understanding of its laws, formulas and SI units is essential not only for scoring marks but also for building the logical reasoning that higher topics such as thermodynamics, electromagnetism and modern physics demand.
This article presents a complete deep-study revision of mechanics. It begins with the basic vocabulary, moves through the core formulas, then discusses the major laws and principles, the way numerical problems are solved, common applications, the pattern of questions asked in examinations and the mistakes that candidates frequently make. Each section is designed so that a reader can revise the entire topic in a single sitting and still retain the key facts.
Basic Terms
- Distance: the total length of the actual path travelled by a body. It is a scalar quantity and is always positive. Its SI unit is the metre (m).
- Displacement: the shortest straight-line distance between the initial and final positions of a body, along with its direction. It is a vector quantity and can be zero, positive or negative.
- Speed: the rate of change of distance with time. It is a scalar quantity, and its SI unit is metre per second (m/s).
- Velocity: the rate of change of displacement with time. It is a vector quantity; average velocity equals total displacement divided by total time.
- Acceleration: the rate of change of velocity with time. It is a vector quantity, and its SI unit is metre per second squared (m/s squared).
- Mass: the quantity of matter contained in a body. It is a scalar and remains constant everywhere in the universe. Its SI unit is the kilogram (kg).
- Force: an external agency that changes, or tends to change, the state of rest or of uniform motion of a body. Its SI unit is the newton (N).
- Momentum: the product of the mass and velocity of a body, given by p = mv. It is a vector quantity, and its SI unit is kilogram metre per second (kg m/s).
- Inertia: the inherent property of a body by virtue of which it resists any change in its state of rest or of uniform motion. It is measured by mass.
Core Concepts and Formulas
- The three equations of motion under uniform acceleration: v = u + at, s = ut + (1/2)at squared, and v squared = u squared + 2as, where u is initial velocity, v is final velocity, a is acceleration, t is time and s is displacement.
- Newton's second law: F = ma, or more generally F = dp/dt, the rate of change of momentum.
- Momentum: p = mv; impulse J = F times t = change in momentum = m(v minus u).
- Weight: W = mg, where g is the acceleration due to gravity, approximately 9.8 m/s squared on the surface of the Earth.
- Work done by a constant force: W = F s cos theta, where theta is the angle between the force and the displacement. Unit: joule (J).
- Kinetic energy: KE = (1/2)mv squared. Work-energy theorem: net work done equals the change in kinetic energy.
- Gravitational potential energy near the Earth's surface: PE = mgh.
- Power: P = W/t, unit watt (W); also P = Fv for a constant force.
- Newton's law of universal gravitation: F = G m1 m2 / r squared, with G = 6.674 times 10 to the power minus 11 N m squared per kg squared.
- Acceleration due to gravity at a planet's surface: g = GM / R squared.
Units and Measurement
Every measurement in mechanics needs a standard. The International System of Units (SI) defines seven base units, of which the metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), mole (amount of substance) and candela (luminous intensity) form the foundation. Mechanics uses the first three most heavily, forming the MKS (metre-kilogram-second) system. Students must also be familiar with the older CGS system (centimetre, gram, second) and the FPS system (foot, pound, second). Dimensional analysis is a powerful tool: by writing a physical quantity in terms of the base dimensions M, L and T, one can check the correctness of an equation, convert units between systems, and even derive relations between quantities.
For example, the dimensional formula of force is M L T to the power minus 2, that of work and energy is M L squared T to the power minus 2, and that of power is M L squared T to the power minus 3. A candidate should be able to recall the dimensional formulas of common quantities instantly, because direct questions on dimensions appear regularly. Accuracy and precision, significant figures, and the use of error analysis (absolute errors add for sums, while fractional errors add for products and powers) round off this essential topic.
Motion in a Straight Line
Kinematics describes motion without asking what causes it. For uniformly accelerated motion in one dimension, the three standard equations listed above solve almost every problem. The key skill is identifying the given quantities and the one that must be found. Free fall is the classic case: with a = g directed downward and neglecting air resistance, a body dropped from rest has u = 0, so its velocity after time t is v = gt and its fall distance is h = (1/2)g t squared.
Graphs are equally important. In a velocity-time graph, the slope gives the acceleration and the area under the curve gives the displacement. In a position-time graph, the slope gives the velocity. Relative velocity, the velocity of one body as observed from another, is obtained by vector subtraction, and it explains problems of two trains crossing, boats crossing rivers, and rain falling on a moving person.
Newton's Laws of Motion
Newton gave three laws that form the foundation of classical mechanics. The first law, the law of inertia, states that a body continues in its state of rest or of uniform motion in a straight line unless it is acted upon by an external force. The second law states that the net force on a body equals the rate of change of its momentum, which gives F = ma when the mass is constant. The third law states that to every action there is an equal and opposite reaction, acting on two different bodies.
From these laws one studies friction (static and kinetic), the motion of bodies connected by strings over pulleys (the Atwood machine), the tension in strings, the normal reaction, and circular motion, where the centripetal force F = mv squared / r is required to keep a body on a curved path. The concept of a free-body diagram, in which every force acting on a single body is drawn, is the most reliable problem-solving technique at this level.
Work, Energy and Power
Work is said to be done when a force produces a displacement. If the force and the displacement are perpendicular, as in carrying a load horizontally, the work done is zero. Positive work increases the kinetic energy of a body, while negative work, such as that done by friction, decreases it. Energy is the capacity to do work; it exists in many forms but is always conserved. The work-energy theorem connects the two ideas: the net work done on a body equals the change in its kinetic energy.
Conservative forces, such as gravity and the spring force, allow potential energy to be defined, whereas friction is a non-conservative force. The principle of conservation of mechanical energy states that, in the absence of non-conservative forces, the sum of the kinetic and potential energies remains constant. Power measures how quickly work is done; a machine of high power can do the same work in less time.
Gravitation
Newton's law of universal gravitation states that 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. The constant of proportionality G is universal and equals about 6.674 times 10 to the power minus 11 N m squared per kg squared. From this law one derives the acceleration due to gravity, g = GM / R squared. Taking the mass of the Earth as about 5.97 times 10 to the power 24 kg and its radius as about 6.37 million metres gives g approximately equal to 9.8 m/s squared.
Important results include the variation of g with altitude (g decreases as height increases), with depth (g decreases and becomes zero at the centre of the Earth), and with latitude (maximum at the poles and minimum at the equator because of the Earth's rotation). Kepler's three laws of planetary motion, which describe elliptical orbits, equal areas swept in equal times, and the relation T squared proportional to a cubed, are natural consequences of gravitation. The escape velocity from the Earth is about 11.2 km/s, and the orbital velocity of a near-Earth satellite is about 7.9 km/s.
Applications in Science and Daily Life
Mechanics is visible everywhere. The safety belts and airbags in vehicles work because of inertia; the recoil of a gun follows the conservation of momentum; a rocket moves forward because it throws hot gases backward. The design of bridges, dams, cranes and machinery rests on the equilibrium of forces and torque. Sports such as cricket, football and the javelin throw are living laboratories of projectile motion. The braking distance of a car depends on kinetic energy and friction. Even the delicate balance of a cyclist taking a turn is an application of centripetal force.
How Numerical Problems Are Solved
A disciplined method saves time. Read the question and list the given quantities with their units. Identify what is asked. Choose the formula that connects the knowns to the unknown. Convert all quantities to SI units before substituting. Draw a free-body diagram whenever forces are involved. Use sign conventions consistently, for example taking upward or forward as positive. Finally, check the dimensions and the reasonableness of the answer. In multiple-choice examinations speed matters, so memorising standard results such as projectile range, time of flight, maximum height and the positions of the centre of mass pays rich dividends.
Examination Trends
In APSC and ADRE general-science papers, mechanics questions are usually factual and formula-based: units of quantities, statements of laws, and simple numericals on speed, acceleration, work, power and gravitation. In NEET and JEE, the questions are more analytical and may combine kinematics with Newton's laws or the conservation of energy. Recent trends show a preference for assertion-reason and matching-type questions, and for conceptual questions about inertia, equilibrium and the distinction between mass and weight. Regular practice of previous years' papers is the best guide to the exact flavour of the questions asked.
Common Mistakes to Avoid
- Confusing distance with displacement, or speed with velocity, and forgetting that displacement can be zero even when the distance travelled is large.
- Using g = 10 m/s squared in one step and 9.8 m/s squared in another within the same problem.
- Forgetting to convert kilometres per hour to metres per second by multiplying by five-eighteenths.
- Treating mass and weight as the same; weight changes with location while mass does not.
- Ignoring the direction of vectors while adding forces or velocities.
- Assuming that work is always done when a force acts, even when there is no displacement or the force is perpendicular to the motion.
- Mixing up the universal gravitational constant G with the acceleration due to gravity g.
- Forgetting that action and reaction act on different bodies and therefore never cancel each other.
How to Prepare This Topic
- Begin by mastering the basic definitions and their SI units, since these form the majority of the easy marks.
- Write all the formulas on a single revision sheet and revise them daily until recall is instant.
- Solve at least twenty numerical problems from each sub-topic, starting with the easy ones and progressing to exam level.
- Practise graph-based and free-body-diagram questions to strengthen conceptual understanding.
- Attempt previous years' question papers under timed conditions and analyse every mistake carefully.
Frequently Asked Questions
What is the difference between mass and weight?
Mass is the quantity of matter in a body, is a scalar, and is the same everywhere in the universe. Weight is the gravitational force acting on the body, given by W = mg, is a vector directed towards the centre of the Earth, and changes from place to place because g changes.
State Newton's second law of motion and give its formula.
The net external force acting on a body is directly proportional to the rate of change of its momentum and acts in the direction of that change. For constant mass it reduces to F = ma, where F is the force in newtons, m is the mass in kilograms and a is the acceleration in metres per second squared.
What is the value of the acceleration due to gravity, and does it change with height?
On the surface of the Earth, g is approximately 9.8 m/s squared. It decreases with increasing height above the surface, and it also decreases with depth below the surface, becoming zero at the centre of the Earth.
Why is the work done by a coolie carrying a load on his head zero?
Because the force he exerts to support the load is upward, while his displacement is horizontal. Since the angle between the force and the displacement is ninety degrees, cos ninety equals zero, and hence W = F s cos theta is zero.
What is escape velocity and what is its value for the Earth?
Escape velocity is the minimum speed with which a body must be projected so that it escapes the gravitational field of a planet and never returns. For the Earth its value is approximately 11.2 kilometres per second.