If the ratio of the weight of a man in a stationary lift and when it is moving downwards with uniform acceleration ‘a’ is 3 : 2, then the value of‘a’ is

1. Newton’s Laws of Motion: The Foundation of Mechanics (basic)

Welcome to your first step in mastering mechanics! To understand how the world moves, we must start with Sir Isaac Newton’s Laws of Motion. These laws are the bedrock of physics, explaining everything from why a ball rolls to how a satellite stays in orbit. At its simplest, motion is governed by Force, which is a push or pull resulting from an object's interaction with another Science, Class VIII NCERT (2025), Exploring Forces, p.77. We measure this force in a unit appropriately named the newton (N) Science, Class VIII NCERT (2025), Exploring Forces, p.65.

Newton’s First Law introduces the concept of Inertia: the tendency of an object to resist changes in its state of motion. If an object is moving in linear motion (a straight line), it wants to keep moving that way at a constant speed unless an external force intervenes Science, Class VII NCERT (2025), Measurement of Time and Motion, p.116. The Second Law gives us the mathematical relationship for this change: Force = mass × acceleration (F = ma). This tells us that the more mass an object has, the more force we need to make it accelerate. Finally, the Third Law reminds us that forces always come in pairs—every action has an equal and opposite reaction.

To differentiate between the types of forces that cause these motions, look at how they interact:

Type of Force	Description	Examples
Contact Forces	Require physical touch between objects.	Muscular force, Frictional force.
Non-contact Forces	Act through a space/field without touching.	Magnetic force, Gravitational force.

Understanding these laws is vital because Force is not just about movement; it can change an object's speed, its direction, or even its shape Science, Class VIII NCERT (2025), Exploring Forces, p.77. As we progress, you will see how these fundamental rules allow us to calculate complex scenarios, like the weight of an object in a moving elevator.

Sources: Science, Class VIII NCERT (2025), Exploring Forces, p.65, 77; Science, Class VII NCERT (2025), Measurement of Time and Motion, p.116

2. Mass vs. Weight: Defining the Constants (basic)

In our daily lives, we often use the words mass and weight as if they mean the same thing. However, in the world of physics and for your UPSC preparation, it is crucial to distinguish between them. Mass is the measure of the actual quantity of matter present in an object Science, Class VIII NCERT (2025), Chapter 9, p. 142. Think of it as the "stuff" an object is made of. Because the amount of matter doesn't change regardless of where you go in the universe, mass is a constant property. Its standard unit is the kilogram (kg).

Weight, on the other hand, is not an inherent property of the object itself, but a measure of the gravitational force with which a planet pulls that object toward its center Science, Class VIII NCERT (2025), Chapter 5, p. 75. Because weight is a force, it is measured in Newtons (N). The relationship is defined by the formula W = mg, where 'm' is mass and 'g' is the acceleration due to gravity. While your mass remains the same on Earth and the Moon, your weight would be much less on the Moon because its gravitational pull is weaker.

Interestingly, weight can even change while you stay on Earth! Gravity is greater near the poles and slightly less at the equator because the Earth is not a perfect sphere; the equator is further from the Earth's center Fundamentals of Physical Geography, Class XI NCERT (2025), Chapter 2, p. 19. Geologists also track gravity anomalies, which are slight variations in gravity caused by the uneven distribution of mass (like heavy mineral deposits) within the Earth's crust Physical Geography by PMF IAS, Earths Interior, p. 58.

Feature	Mass	Weight
Definition	Quantity of matter in an object.	Force of gravity acting on an object.
Constancy	Constant everywhere.	Changes with location/gravity.
SI Unit	Kilogram (kg)	Newton (N)
Measurement	Measured using a beam balance.	Measured using a spring balance.

Sources: Science, Class VIII NCERT (2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.142; Science, Class VIII NCERT (2025), Chapter 5: Exploring Forces, p.75; Fundamentals of Physical Geography, Class XI NCERT (2025), Chapter 2: The Origin and Evolution of the Earth, p.19; Physical Geography by PMF IAS, Earths Interior, p.58

3. Free Body Diagrams and Normal Reaction Force (intermediate)

To master mechanics, we must first learn to visualize the invisible. A Free Body Diagram (FBD) is a graphical tool used to isolate an object and represent all the forces acting upon it as vectors. Just as ray diagrams allow us to study the nature and position of images in optics Science, Class X, Light – Reflection and Refraction, p. 153, an FBD allows us to translate a physical situation into a mathematical equation. When drawing an FBD, we represent the object as a single point and draw arrows pointing away from it to show the direction and magnitude of forces like gravity (weight) and contact forces.

One of the most critical contact forces is the Normal Reaction Force (N). This force arises whenever two surfaces are in physical contact Science, Class VIII, Exploring Forces, p. 66. It is called "normal" because it always acts perpendicular to the surface of contact. Crucially, what we perceive as our "weight" when standing on a scale is not actually the pull of gravity itself, but the Normal Force the scale exerts back on us. If the surface you are standing on accelerates, this Normal Force changes, leading to the phenomenon of apparent weight.

Consider a person of mass m in a lift. When the lift is stationary, the upward Normal Force exactly balances the downward force of gravity (weight), so N = mg. However, if the lift accelerates downward with acceleration a, the net force must be downward. According to Newton's Second Law (F = ma), the equation becomes mg - N = ma, which rearranges to N = m(g - a). This explains why you feel lighter when a lift starts to descend; the floor doesn't need to push up as hard because you are "falling" along with it.

Lift State	Net Force Equation	Apparent Weight (N)	Sensation
Stationary or Constant Velocity	N - mg = 0	N = mg	Normal Weight
Accelerating Upwards (a)	N - mg = ma	N = m(g + a)	Heavier
Accelerating Downwards (a)	mg - N = ma	N = m(g - a)	Lighter

Sources: Science, Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.153; Science, Class VIII, NCERT (Revised ed 2025), Exploring Forces, p.66

4. Connected Concept: Variations in Gravitational Acceleration (intermediate)

In our study of mechanics, we often treat the acceleration due to gravity (g) as a constant 9.8 m/s². However, for a UPSC aspirant, it is vital to understand that g is a variable influenced by geography and motion. Firstly, gravity depends on your distance from the Earth's center. Because the Earth is an oblate spheroid—bulging at the equator and flattened at the poles—you are technically closer to the Earth's center at the poles. Consequently, gravity is stronger at the poles and weaker at the equator. This shape is partially due to the Earth's rotation Physical Geography by PMF IAS, Earths Interior, p. 59. Similarly, as you ascend to high altitudes, such as the Mana Pass (5,611 m) or Thang La (5,359 m) in the Himalayas, you move further from the Earth's mass, causing g to decrease slightly Geography of India, Majid Husain, Physiography, p. 21-22.

Beyond geographical location, we must distinguish between True Weight and Apparent Weight. True weight is the actual gravitational pull exerted on you (W = mg). Apparent weight, however, is the normal force (the support force) you feel from the surface you are standing on. This is most famously demonstrated in a moving lift (elevator). When a lift is stationary or moving at a constant speed, your apparent weight equals your true weight. But if the lift accelerates downwards at a rate 'a', the floor moves away from you, reducing the support force. Your apparent weight (N) becomes N = m(g - a) Science, Class VIII NCERT (Revised ed 2025), Chapter 5, p. 72. This is why you feel a "sinking" sensation in your stomach when a lift starts its descent.

Conversely, if the lift accelerates upwards, the floor pushes against you more forcefully to overcome both gravity and the required upward acceleration. In this case, your apparent weight increases to N = m(g + a). If a lift were to go into 'free fall' (where a = g), the formula N = m(g - g) shows that your apparent weight would become zero, resulting in a state of weightlessness, even though the Earth's gravity is still pulling on you just as strongly as before.

Sources: Science, Class VIII NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.72; Physical Geography by PMF IAS, Earths Interior, p.59; Geography of India, Majid Husain, Physiography, p.21-22

5. Connected Concept: Friction and Terminal Velocity (exam-level)

To understand Terminal Velocity, we must first look at its precursor: Friction. Friction is a contact force that arises because no surface is perfectly smooth. Even those that appear polished have microscopic irregularities that "lock" into each other when surfaces interact, opposing any attempt at movement Science, Class VIII . NCERT (Revised ed 2025), Chapter 5, p. 68. While we often think of friction only between solids, it is equally present in fluids like air and water. In geography, for instance, the irregularities of the Earth's surface resist wind movement, slowing it down significantly compared to the smoother surface of the sea Physical Geography by PMF IAS, Pressure Systems and Wind System, p. 307.

When an object falls through the atmosphere, it experiences a tug-of-war between two opposing forces. Gravity (its weight) pulls it downward, while Air Resistance (a form of fluid friction) pushes upward. Initially, gravity is much stronger, so the object accelerates. However, a unique property of fluid friction is that it increases as the object's speed increases. The faster you fall, the harder the air pushes back against you.

Eventually, a state of dynamic equilibrium is reached. This occurs when the upward force of friction grows large enough to exactly equal the downward force of gravity. At this precise moment, the net force acting on the object becomes zero (F_net = 0). According to Newton’s laws, if there is no net force, there is no acceleration. The object stops speeding up and continues to fall at a constant, maximum speed known as Terminal Velocity.

Phase of Fall	Force Comparison	Motion Status
Initial Drop	Weight > Air Resistance	Accelerating Downward
Middle Phase	Weight > Air Resistance (but gap is closing)	Increasing speed at a slower rate
Terminal Velocity	Weight = Air Resistance	Constant Speed (Zero Acceleration)

Sources: Science, Class VIII . NCERT (Revised ed 2025), Exploring Forces, p.68; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.307

6. Apparent Weight and Pseudo-Forces in a Lift (exam-level)

To understand why you feel 'weightless' for a split second when a lift starts moving downwards, we must first distinguish between True Weight and Apparent Weight. True weight is simply the gravitational force with which the Earth pulls you downward, calculated as W = mg Science, Class VIII. NCERT (Revised ed 2025), Chapter 5, p.72. However, your 'feeling' of weight doesn't come from gravity itself, but from the Normal Reaction Force (N) exerted by the floor against your feet. In a stationary lift, these two forces are equal (N = mg), so you feel your normal weight. Everything changes when the lift begins to accelerate. According to Newton's Second Law (F = ma), if the lift accelerates downwards at a rate 'a', the net force acting on you must be 'ma' in the downward direction. This means the downward pull of gravity (mg) is now stronger than the upward push of the floor (N). The equation becomes mg - N = ma, which simplifies to N = m(g - a). Because the floor is effectively 'dropping away' from you, it doesn't need to push back as hard, making your Apparent Weight (N) less than your True Weight (mg). Conversely, if the lift accelerates upward, the floor must push harder to overcome gravity and provide upward momentum, leading to N = m(g + a), making you feel heavier.

Lift Motion	Formula for Apparent Weight (N)	Sensation
Stationary or Constant Velocity	N = mg	Normal weight
Accelerating Downwards (a)	N = m(g - a)	Lighter
Accelerating Upwards (a)	N = m(g + a)	Heavier
Free Fall (a = g)	N = m(g - g) = 0	Weightlessness

In competitive exams, you will often encounter problems where the weight changes by a certain ratio. For example, if your apparent weight becomes two-thirds of your true weight while descending, you can set up the equation mg / m(g - a) = 3 / 2. Solving this simple algebraic relation reveals that the lift's acceleration (a) must be exactly one-third of gravity (g/3). Understanding this balance between gravity and acceleration is the key to mastering mechanics in non-inertial frames.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 5: Exploring Forces, p.72

7. Solving the Original PYQ (exam-level)

Great job mastering the fundamental forces! This question brings together your understanding of Newton’s Second Law and the concept of apparent weight. In a stationary lift, the scale measures your true weight (mg) because the normal force and gravity are in equilibrium. However, when the lift accelerates downwards, the floor "falls away" from you slightly, reducing the contact force. As you learned in Science, Class VIII. NCERT (Revised ed 2025), the apparent weight (N) during downward acceleration is calculated as m(g - a).

To arrive at the correct answer, we set up the ratio provided in the question: 3/2 = mg / m(g - a). By canceling the mass (m) from both sides, the relationship simplifies to 3(g - a) = 2g. Expanding this gives 3g - 3a = 2g, which rearranges to 3a = g. Thus, the acceleration a = g/3, making (B) the correct choice. This logical progression shows how a simple change in the frame of reference (the accelerating lift) alters the weight perceived by the observer.

UPSC often includes options like (D) 2g/3 as a trap for students who might confuse the final ratio of the weights with the value of acceleration itself. Similarly, (C) g represents the scenario of free fall, where the apparent weight would be zero, not two-thirds. Always double-check your algebraic signs—if the lift were moving upward, the formula would have been m(g + a), which would make the man feel heavier. Mastering these nuances is key to avoiding the common pitfalls in the Prelims.