1.0 Introduction to Force
In our previous chapter, we studied how objects move. Now, we explore why they move. A Force is a push or pull upon an object resulting from the object's interaction with another object. Force can change the state of rest or motion, the speed, the direction, and even the shape of a body.
Contact vs. Non-Contact Forces
- Contact Forces: Forces that act only when objects are in physical touch (e.g., Frictional force, Normal reaction force, Tension).
- Non-Contact Forces: Forces that act through a space without physical contact (e.g., Gravitational force, Electrostatic force, Magnetic force).
1.1 Newton's First Law: The Law of Inertia
Newton's First Law states: "An object remains in its state of rest or of uniform motion in a straight line unless compelled to change that state by an applied external force." This property of a body to resist any change in its state is called Inertia.
Measurement of Inertia
Inertia is a quantitative property. The more mass a body has, the greater is its inertia.
$$\text{Inertia} \propto \text{Mass}$$
Types of Inertia
- Inertia of Rest: Tendency to remain at rest (e.g., passengers fall backward when a bus starts suddenly).
- Inertia of Motion: Tendency to maintain uniform motion (e.g., passengers lean forward when brakes are applied).
- Inertia of Direction: Tendency to maintain direction (e.g., passengers are thrown outwards when a car takes a sharp turn).
When explaining inertia examples in exams, always mention two parts:
1. The part of the body in contact with the vehicle changes its state immediately.
2. The upper part of the body tries to maintain its original state due to Inertia.
A truck and a car are both at rest. Which one requires more force to start moving and why?
Solution:
1. A truck has much greater mass than a car.
2. Since Inertia is directly proportional to mass, the truck has higher inertia of rest.
3. Therefore, it offers more resistance to change and requires a larger force to be set into motion.
Final Answer: The truck, due to its greater mass/inertia.
Isaac Newton didn't "invent" the First Law. It was actually Galileo who first proposed the concept of inertia by observing that a ball rolling on a perfectly smooth horizontal plane would never stop!
2.0 Momentum and Newton's Second Law
Newton's First Law tells us what happens in the absence of force. The Second Law provides a mathematical way to measure force. Before we dive in, we must understand Linear Momentum, which is the "quantity of motion" contained in a body.
Linear Momentum ($p$)
It is defined as the product of the mass of a body and its velocity.
- Type: Vector quantity (direction is the same as velocity).
- S.I. Unit: $kg \cdot m/s$.
- Formula: $\vec{p} = m \cdot \vec{v}$
2.1 Newton's Second Law of Motion
Newton's Second Law states: "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."
Mathematical Form
$$F = m \times a$$
Where: $F$ = Force (Newtons), $m$ = Mass, $a$ = Acceleration.
Derivation Hint: $F \propto \frac{\Delta p}{t} \Rightarrow F \propto \frac{m(v-u)}{t} \Rightarrow F \propto ma$.
2.2 Force Units and Impulse
The S.I. unit of force is the Newton (N). One Newton is the force which produces an acceleration of $1\,m/s^2$ in a body of mass $1\,kg$.
- CGS Unit: dyne ($1\,N = 10^5$ dyne).
- Gravitational Unit: $kgf$ (kilogram-force). $1\,kgf \approx 9.8\,N$.
A large force acting for a very short time is called an Impulsive Force.
Impulse = Force $\times$ Time = Change in Momentum.
This is why a cricket player pulls his hands back while catching a ball—to increase the time ($t$), which reduces the force ($F$) exerted on his hands.
A force of $5\,N$ acts on a body of mass $2\,kg$ for $4\,s$. Calculate (a) acceleration and (b) change in momentum.
Solution:
1. Acceleration ($a$): $F = ma \Rightarrow a = F/m = 5/2 = \mathbf{2.5\,m/s^2}$.
2. Change in Momentum ($\Delta p$): $\Delta p = F \times t = 5 \times 4 = \mathbf{20\,kg \cdot m/s}$.
Final Answer: Acceleration is $2.5\,m/s^2$ and change in momentum is $20\,kg \cdot m/s$.
The Newton-second ($N \cdot s$) is an alternative unit for momentum. Even though the units $kg \cdot m/s$ and $N \cdot s$ look different, they are mathematically identical!
3.0 Newton's Third Law of Motion
Newton’s Third Law explains the nature of interaction between two bodies. It states: "To every action, there is always an equal and opposite reaction." It is important to remember that action and reaction always act on two different bodies simultaneously.
Key Features of Third Law
- Action and reaction occur in pairs.
- They are equal in magnitude but opposite in direction.
- Since they act on different bodies, they never cancel each other out.
Examples in Daily Life
- Walking: We push the ground backward (Action), and the ground pushes us forward (Reaction).
- Recoil of a Gun: When a bullet is fired forward (Action), the gun moves backward (Reaction).
- Swimming: The swimmer pushes water backward (Action), and water pushes the swimmer forward (Reaction).
3.1 Newton's Universal Law of Gravitation
Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Universal Gravitational Formula
$$F = G \frac{m_1 m_2}{r^2}$$
Where: $G$ = Universal Gravitational Constant ($6.67 \times 10^{-11}\,N \cdot m^2/kg^2$)
Never confuse these two!
Mass: Quantity of matter (Scalar, remains constant everywhere).
Weight ($W = mg$): Force of gravity on an object (Vector, changes with location as $g$ changes).
A man weighs 600 N on Earth. What will be his (a) mass and (b) weight on the Moon? (Take $g_{earth} = 10\,m/s^2$ and $g_{moon} = \frac{1}{6}g_{earth}$)
Solution:
1. Mass on Earth: $m = W/g = 600/10 = \mathbf{60\,kg}$. Mass remains same on Moon.
2. Weight on Moon: $W_{moon} = m \times g_{moon} = 60 \times (10/6) = \mathbf{100\,N}$.
Final Answer: Mass = $60\,kg$; Weight on Moon = $100\,N$.
Gravitational force is the weakest force in nature, yet it controls the motion of stars and galaxies! Its value is so small that we don't feel the attraction between everyday objects like two chairs or two people.
4.0 Free Fall and Weightlessness
When an object moves solely under the influence of the Earth's gravitational pull, it is said to be in Free Fall. During this state, the object experiences a constant acceleration ($g$) regardless of its mass, provided air resistance is neglected.
Understanding Weightlessness
Weightlessness is a sensation where an individual feels as if they have no weight. It occurs when there is no reaction force (normal force) acting on the body from the surface it is in contact with.
- Apparent Weight: The force a body exerts on the surface supporting it.
- True Weight ($W = mg$): The actual gravitational pull of the Earth.
4.1 The Case of a Falling Lift
The sensation of weight changes depending on the acceleration of the platform (like a lift) you are standing on:
| Condition | Apparent Weight ($R$) | Sensation |
|---|---|---|
| At Rest / Uniform Velocity | $R = mg$ | Normal Weight |
| Accelerating Upwards ($a$) | $R = m(g + a)$ | Feeling Heavier |
| Accelerating Downwards ($a$) | $R = m(g - a)$ | Feeling Lighter |
| Free Fall ($a = g$) | $R = 0$ | Weightlessness |
A common misconception is that gravity is zero in space. In an orbiting satellite, astronauts feel weightless not because there is "no gravity," but because both the astronaut and the satellite are in a state of constant free fall towards the Earth.
Equations for Free Fall
Since $u=0$ and $a=g$, the motion equations become:
$$v = gt \quad | \quad h = \frac{1}{2}gt^2 \quad | \quad v^2 = 2gh$$
A boy of mass $50\,kg$ stands in a lift which is moving down with an acceleration of $2\,m/s^2$. What is his apparent weight? (Take $g = 10\,m/s^2$)
Solution:
1. Mass ($m$): $50\,kg$.
2. Acceleration ($a$): $2\,m/s^2$ (Downwards).
3. Formula: $R = m(g - a)$.
4. Calculation: $R = 50(10 - 2) = 50 \times 8 = 400\,N$.
Final Answer: The apparent weight is $400\,N$ (which is $100\,N$ less than his true weight).
The "Vomit Comet" is a specially designed NASA airplane that flies in parabolic arcs. At the top of the arc, the plane enters a controlled free-fall, allowing astronauts to practice moving in a weightless environment for about 25 seconds at a time!