9. Force and Laws of Motion | connecting-the-dots

1. Concept of Force

Definition: A force is a push, hit, or pull on an object that changes its state of motion or rest, or changes its shape and size.
Effect of Force:
- Put a stationary object into motion.
- Stop a moving object.
- Change the magnitude of the velocity (speed up or slow down).
- Change the direction of motion.
- Change the shape and size of objects.
Observation: Force is not seen, tasted, or felt directly, but its effects are observed.

Feature	Balanced Forces	Unbalanced Force
Effect on Motion	Do not change the state of rest or motion of an object.	Brings an object into motion or changes its speed/direction (accelerates it).
Net Force	Net force acting on the object is zero.	Net force acting on the object is non-zero.
Example	Pulling a block from both sides with equal force; friction force balancing a small push.	Pushing a box harder than the friction force; gravity acting on a falling marble.

Friction: A force that opposes motion and acts between two surfaces in contact.

Statement: An object remains in a state of rest or of uniform motion in a straight line unless compelled to change that state by an applied (unbalanced) force.
Inertia: The natural tendency of an object to resist a change in its state of rest or of uniform motion.
Mass and Inertia: The mass of an object is a measure of its inertia. Heavier/more massive objects offer larger inertia.
Real-life Examples:
- Rest: Tendency to fall backwards when a bus suddenly starts (body resists motion).
- Motion: Tendency to fall forwards when a moving car brakes suddenly (body resists stopping).
- Sharp turn: Tendency to be thrown sideways (body resists changing its straight-line motion).

Concept: A quantity of importance that combines an object's mass and its velocity. The impact produced by an object depends on its mass and velocity.
Formula: $p = m \times v$ $p = m \times v$
- $m$ : mass
- $v$ : velocity
Nature: It has both magnitude and direction (same direction as velocity).
SI Unit: kilogram-metre per second ( $\text{kg m s}^{-1}$ ).
Force and Momentum: The application of an unbalanced force produces a change in momentum.

Statement: The rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of force.
Mathematical Formulation:
- Force $F \propto \text{Rate of change of momentum}$
- $F \propto \frac{(mv - mu)}{t} \implies F = kma$
- By defining one unit of force as the amount that produces $1 , \text{m s}^{-2}$ acceleration in $1 , \text{kg}$ mass, the constant $k$ becomes 1.
- Final Formula: $\mathbf{F = ma}$
Unit of Force:
- $\text{kg m s}^{-2}$ , also called newton ( $\mathbf{N}$ ).
- $1 , \text{N}$ is the force that produces an acceleration of $1 , \text{m s}^{-2}$ on an object of $1 , \text{kg}$ mass.
Relating First and Second Law: If $F=0$ , then $a=0$ . This means $v=u$ , so the object continues to move with uniform velocity $u$ or remains at rest if $u=0$ .
Applications:
- A cricket fielder pulls hands backwards while catching a ball to increase the time over which momentum changes, thereby decreasing the force felt.
- High jump athletes fall on a cushioned/sand bed to increase the time of stopping, decreasing the impact force.

Statement: When one object exerts a force on another object (action), the second object instantaneously exerts a force back on the first (reaction).
- Alternative Statement: To every action, there is an equal and opposite reaction.
Key Points:
- The action and reaction forces are always equal in magnitude but opposite in direction.
- They always act on two different objects, never on the same object.
Examples:
- Walking: You push the road backward (action), and the road pushes you forward (reaction).
- Recoil of a Gun: The gun exerts a forward force on the bullet, and the bullet exerts an equal and opposite backward force on the gun (recoil).
- Sailor Jumping: Sailor pushes the boat backward, and the boat pushes the sailor forward.

Statement: The sum of momenta of two objects before collision is equal to the sum of momenta after the collision, provided there is no external unbalanced force acting on them. The total momentum of an isolated system remains unchanged or conserved.
Mathematical Derivation: Based on the Third Law of Motion ( $F_{AB} = -F_{BA}$ $F_{A B} = - F_{B A}$ ).
- Formula: $m_A u_A + m_B u_B = m_A v_A + m_B v_B$ $m_{A} u_{A} + m_{B} u_{B} = m_{A} v_{A} + m_{B} v_{B}$
  - $m_A, m_B$ : masses of object A and B
  - $u_A, u_B$ : initial velocities
  - $v_A, v_B$ : final velocities
Applications: Recoil of a gun, motion of rockets, collisions.

Fundamental laws in physics (like momentum, energy, charge).
Based on observations and experiments.
A conservation law cannot be proved; it is verified or disproved by experiments.
Conservation of Energy
Conservation of Angular Momentum
Conservation of Charge