⚡ Fast Revision: Force & Newton's First Law
- Force: An external agency (push or pull) that changes or tends to change the state of rest, uniform motion, or the direction/dimensions of a body.
- Inertia: The inherent property of an object by virtue of which it resists any change to its state of rest or uniform motion in a straight line.
- Mass as a Measure: Inertia is not a physical quantity; it depends entirely on mass. A heavier body possesses greater inertia than a lighter body.
Force ($F$): SI Unit: Newton ($\text{N}$) | CGS Unit: Dyne ($\text{dyn}$)
Gravitational Unit: Kilogram-force ($\text{kgf}$) where $1\text{ kgf} = g\text{ N} \approx 9.8\text{ N}$
$1\text{ N} = 10^5\text{ dyn}$
Types of Inertia
| Type of Inertia | Definition Statement | Classic Exam Example |
|---|---|---|
| Inertia of Rest | Tendency of a body to remain at rest. | Passengers fall backward when a stationary bus starts suddenly. |
| Inertia of Motion | Tendency of a body to maintain its uniform speed. | Passengers lean forward when a moving bus applies brakes suddenly. |
| Inertia of Direction | Tendency of a body to maintain its straight-line path. | An umbrella protects from rain because water drops cannot change direction on their own. |
Thinking that Newton's First Law defines the quantitative magnitude of force. Fix: The First Law provides only the qualitative definition of force and describes inertia. Quantitative measurement comes strictly from the Second Law.
⚡ Fast Revision: Momentum & Newton's Second Law
- Linear Momentum ($p$): The measure of the total quantity of motion contained within a moving body, determined by the product of its mass and velocity.
- Newton's Second Law: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.
- Force Quantification: Since $F \propto \frac{\Delta p}{t}$, substituting $p = mv$ under constant mass yields the foundational equation $F = ma$.
Linear Momentum ($p$): SI Unit: $\text{kg m s}^{-1}$ | CGS Unit: $\text{g cm s}^{-1}$
$$\mathbf{p = m \cdot v}$$
$$\mathbf{F = \frac{\Delta p}{\Delta t} = \frac{m(v - u)}{t} = m \cdot a}$$
Proportionality of Force Relationships
| Fixed Variable | Proportionality | Practical Interpretation |
|---|---|---|
| If Mass ($m$) is constant | $F \propto a$ | To double the acceleration of a fixed mass, you must exert twice the force. |
| If Force ($F$) is constant | $a \propto \frac{1}{m}$ | The same force applied to a heavier body produces smaller acceleration. |
Forgetting that momentum is a vector quantity during direction reversals. Fix: If a ball of mass $m$ strikes a wall with velocity $v$ and rebounds with the same speed, the change in momentum is not zero; it is $\Delta p = -mv - mv = -2mv$.
⚡ Fast Revision: Second Law Applications & Newton's Third Law
- Impact Time Manipulation: According to $F = \frac{\Delta p}{t}$, increasing the duration ($t$) over which momentum drops to zero significantly reduces the damaging impact force ($F$).
- Newton's Third Law: To every action, there is always an equal and opposite reaction. Action and reaction forces act on different bodies simultaneously.
- Simultaneous Nature: A single isolated force cannot exist in nature. Forces always occur in matched action-reaction pairs ($F_{\text{AB}} = -F_{\text{BA}}$).
$$\mathbf{F_{\text{action}} = -F_{\text{reaction}}}$$
The negative sign strictly signifies opposing vector directions.
Everyday Mechanical Phenomenon Explained
| Observed Event | Governing Physical Law | Underlying Mechanism |
|---|---|---|
| Cricketer pulling hands back | Newton's Second Law ($F \propto \frac{1}{t}$) | Increases catching time to minimize the stopping force, preventing injury. |
| Recoil of a heavy gun | Newton's Third Law ($F_{\text{AB}} = -F_{\text{BA}}$) | The bullet is driven forward (Action); the gun is kicked backward (Reaction). |
| Athlete jumping on sand/cushion | Newton's Second Law ($\Delta t \text{ extension}$) | The soft surface yields slowly, reducing the ground reaction force on the feet. |
Thinking action and reaction forces cancel each other out to create static equilibrium. Fix: Action and reaction forces never cancel each other because they act on two entirely different objects. Cancellation only happens when equal and opposite forces target a single body.
⚡ Fast Revision: Gravitation, Mass & Weight
- Newton's Law of Gravitation: Every particle 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 them.
- Universal Gravitational Constant ($G$): A scalar constant whose value remains identical everywhere in the universe, independent of the medium separating the masses.
- Mass vs Weight: Mass is the measure of matter contained in a body (invariant scalar), whereas Weight is the force with which Earth attracts that body ($W=mg$, variable vector).
Universal Constant ($G$): Value: $6.67 \times 10^{-11}$ | SI Unit: $\text{N m}^2\text{ kg}^{-2}$
Acceleration due to gravity ($g$): Avg Value: $9.8\text{ m s}^{-2}$ | SI Unit: $\text{m s}^{-2}$
$$F = G\frac{m_1 m_2}{r^2}$$
$$g = \frac{G \cdot M}{R^2}$$
Mass vs Weight Comparison Matrix
| Characteristic | Mass | Weight |
|---|---|---|
| Core Definition | Quantity of matter contained in a body. | Force exerted by gravity pulling the body. |
| Measurement Device | Beam Balance / Common Balance. | Spring Balance. |
| Value at Earth's Center | Remains constant ($\gt 0$). | Becomes exactly zero (since $g = 0$). |
Using the terms $G$ and $g$ interchangeably in numerical problems. Fix: $G$ is a universal constant value everywhere. $g$ changes with location, decreasing at high altitudes, deep mines, and dropping to zero at the planetary center or in deep interstellar space.
⚡ Fast Revision: Inverse-Square Law & Variations in g
- Inverse Square Relationship: The gravitational pull between two point masses changes inversely with the square of the distance ($F \propto \frac{1}{r^2}$). If distance is doubled, the force decreases to one-fourth.
- Shape of the Earth Effect: Earth is an oblate spheroid. The equatorial radius is larger than the polar radius by roughly $21\text{ km}$. Consequently, $g$ is maximum at poles and minimum at the equator.
- Weight Variation Rule: Because $W = mg$, an individual's mass stays identical everywhere, but their weight scales directly with the local value of $g$ as they move across locations.
$$g_{\text{poles}} \gt g_{\text{equator}}$$
$$W_{\text{moon}} = \frac{1}{6} W_{\text{earth}} \quad \left(\text{since } g_{\text{moon}} \approx \frac{g_{\text{earth}}}{6}\right)$$
(Hyperbolic Curve)
Exam Proportionality Matrix
| Change Parameters | Mathematical Calculation | Resulting Force Impact |
|---|---|---|
| Distance between masses is doubled ($2r$) | $F' \propto \frac{1}{(2r)^2} = \frac{1}{4r^2}$ | Force drops to $\frac{1}{4}$th of initial value. |
| Distance between masses is halved ($\frac{r}{2}$) | $F' \propto \frac{1}{(r/2)^2} = \frac{4}{r^2}$ | Force increases to $4$ times its initial value. |
| One of the interacting masses is doubled | $F' \propto (2m_1) \cdot m_2$ | Force is exactly doubled. |
Assuming that a body's mass drops to zero when placed at the center of the Earth because it becomes weightless. Fix: Weight drops to zero because $g = 0$ at the Earth's center, but mass is invariant and remains unchanged.