Notes on Newton's First Law, Net External Force, Inertia, and Illustrative Examples (Chapter 4)

Newton's First Law (inertia) states that if an object is unaccelerated (either at rest or moving with uniform velocity in a straight line), the net external force on it must be zero: $\sum \mathbf{F}_{\text{ext}} = 0.$
This does not require that every individual force is zero; rather, the external forces present must cancel so that their vector sum is zero.
Gravity is everywhere; on Earth, every object experiences gravitational force due to the Earth (weight) in addition to other forces such as friction, viscous drag, etc.
When an object is at rest or in uniform straight-line motion, it is not because there are no forces, but because the external forces cancel out to give zero net external force.

Scenario: A book at rest on a horizontal surface.
External forces: weight W acting downward and the normal force R from the table acting upward.
R is a self-adjusting force: it adjusts its magnitude to balance the weight when in contact with the surface.
Correct balance condition (equilibrium): the net external force is zero, so the upward normal balances the downward weight:
- Magnitudes are equal: $R = W = mg$
- Directions are opposite: R upward, W downward.
Important clarification: It is incorrect to say simply "W = R, so the book is at rest" as the justification. The proper statement is: "Since the book is observed to be at rest, the net external force is zero (by the First Law). This implies that the normal force must be equal and opposite to the weight, i.e. $R = W.$ "
Reference to Fig. 4.2(a): depicts a book at rest with forces W downward and R upward.

Two stages are highlighted: acceleration from rest, and motion with uniform velocity.
When the car starts from rest and speeds up (Fig. 4.2(b) in the text), there must be a net external force along the road to produce acceleration; this force cannot come from internal interactions within the car and must come from an external source.
The external force along the road is friction between the tires and the road. This friction provides the net external force that accelerates the car as a whole.
As the car reaches a steady (constant) velocity, the net external force on the car becomes zero, yielding no further acceleration.
Note: Friction is discussed as a key external force along the road; more details on friction are covered in section 4.9.
Basic relation (Newton's second law): along the road, if the friction is the only external force contributing to the acceleration,
- The net force: $F<em>{\text{net}} = F</em>{\text{friction}}$
- The acceleration: $a = \frac{F<em>{\text{net}}}{m} = \frac{F</em>{\text{friction}}}{m}$
The acceleration cannot be attributed to internal forces within the car; friction with the road is essential for the acceleration.

The bus starts abruptly and we experience a jerk: we are thrown backward relative to the bus.
Explanation in terms of inertia and friction:
- Our feet are in contact with the floor. If acceleration is moderate, static friction between feet and floor is sufficient to accelerate our feet along with the bus.
- The rest of the body tends to stay in its state of motion due to inertia (the body is not perfectly rigid and allows some internal deformation).
- As the feet accelerate with the floor, muscular forces act to move the rest of the body along with the bus.
When the bus stops suddenly:
- The feet stop due to friction that prevents relative motion between feet and bus floor.
- The rest of the body tends to continue moving forward due to inertia.
- Restoring muscular forces again act to bring the body to rest with the bus.
This illustrates how inertia, friction, and internal muscular responses cooperate to produce coordinated motion with a moving surface.

Problem setup: An astronaut is accelerating with a spaceship in interstellar space at a constant rate of $100\,\mathrm{m\,s^{-2}}$ . The astronaut separates from the spaceship. What is the astronaut's acceleration immediately after separation? (Assume no nearby stars to exert gravitational force on him.)
Key assumptions:
- No nearby stars -> negligible gravitational field acting on the astronaut.
- The spaceship's own gravity is negligible on the astronaut after separation.
Reasoning:
- After separation, the astronaut experiences no significant external forces (net external force is zero): $\sum \mathbf{F}_{\text{ext}} = 0.$
- By Newton's second law, the acceleration is $\mathbf{a} = \frac{\mathbf{F}_{\text{net}}}{m} = 0.$
- Therefore, the astronaut's acceleration immediately after separation is $a = 0$ (the astronaut continues with the instantaneous velocity at separation, but with zero further acceleration until another force acts).
Important takeaway: Even though the spaceship was accelerating before separation, once detached in empty space with negligible external forces, the astronaut experiences zero net force and hence zero acceleration at the instant after separation.

The distinction between the presence of forces and the net external force: forces can exist but cancel to yield zero net force, producing no acceleration.
The role of self-adjusting normal force: the normal force adjusts to whatever is needed to balance the weight when contact is maintained between surfaces.
Friction as an external force: friction is often the mechanism by which external forces act to change the motion of a system (e.g., car acceleration) and is distinct from internal forces within a body.
Inertia as a property of matter: objects resist changes to their state of motion; this is manifested in everyday experiences such as starting and stopping vehicles, or objects in free space after separation.
Real-world relevance: understanding these principles is crucial for interpreting everyday phenomena (driving, sitting in a moving vehicle, walking) and for analyzing systems where external forces act in different directions, including space scenarios without significant gravitational forces.

Weight: $W = mg$ (downward on Earth)
Normal force: $R$ (upward), with equilibrium condition for a stationary book: $R = W = mg$
Net external force and acceleration: $\sum \mathbf{F}_{\text{ext}} = m\mathbf{a}$
In equilibrium along a surface: movement occurs with zero net force along that direction; outside that direction, external forces may be nonzero.
Instantaneous acceleration in space example: $a = 0$ when no external forces act on the astronaut after separation.

An object at rest or in uniform motion has zero net external force, not necessarily zero individual forces.
The normal force can balance weight; this balance is described by the First Law rather than by a claim that forces simply cancel.
Friction is a key external force that can cause acceleration (as in a car starting from rest); it is not an internal force.
Inertial effects explain why a person feels thrown when a vehicle starts or stops; real-world body responses involve both friction and muscular action.
In space, without external forces, separation events lead to zero acceleration immediately after separation, even if the system was accelerating previously.