Dynamics: Newton’s Laws of Motion – Structured Study Notes

Chapter 4: Dynamics – Newton’s Laws of Motion

4.1 Force

In physics, a force is defined as a push or pull that can change the motion of an object. Forces are fundamental to understanding how objects interact and move.

• Definition: A force is any interaction that, when unopposed, will change the motion of an object.

• Measurement: The magnitude of a force can be measured using a spring scale.

• Effect: An object at rest requires a force to start moving; a moving object needs a force to change its velocity (speed or direction).

4.2 Newton’s First Law of Motion (Law of Inertia)

Newton’s First Law states the conditions under which an object’s motion remains unchanged. It introduces the concept of inertia and inertial reference frames.

• Statement: Every object continues in its state of rest, or of uniform velocity in a straight line, as long as no net force acts on it.

• Inertia: The tendency of an object to resist changes in its motion.

• Inertial Reference Frame: A frame of reference in which Newton’s first law holds true. Rotating or accelerating frames are not inertial.

4.3 Mass

Mass is a measure of an object’s inertia and is a fundamental property in dynamics. It is distinct from weight.

• Definition: Mass is the measure of inertia of an object; it quantifies how much an object resists acceleration.

• SI Unit: Kilogram (kg).

• Difference from Weight: Mass is intrinsic to the object, while weight is the force exerted by gravity on that mass.

• Example: On the Moon, your weight is less due to lower gravity, but your mass remains unchanged.

4.4 Newton’s Second Law of Motion

Newton’s Second Law establishes the quantitative relationship between force, mass, and acceleration.

• Statement: The acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

• Equation:

$\Sigma F = m a$

• Vector Nature: Force and acceleration are vectors; the law applies to each coordinate axis.

• SI Unit of Force: Newton (N), where $1\,\text{N} = 1\,\text{kg} \cdot \text{m}/\text{s}^2$.

• Pound: A unit of force, not mass; can be converted to newtons.

Table: Units for Mass and Force

Quantity	SI Unit	Other Unit
Mass	kilogram (kg)	gram (g)
Force	newton (N)	pound (lb)
Additional info:	1 N = 1 kg·m/s²	1 lb ≈ 4.448 N

4.5 Newton’s Third Law of Motion

Newton’s Third Law describes the mutual nature of forces between two objects.

• Statement: Whenever one object exerts a force on a second object, the second exerts an equal force in the opposite direction on the first.

• Action-Reaction: Forces always come in pairs, acting on different objects.

• Notation: Subscripts indicate the object acted upon and the source of the force (e.g., $F_{AB}$ is the force on A by B).

• Example: Rocket propulsion: The rocket expels gases backward, and the reaction force propels the rocket forward.

$F_{AB} = -F_{BA}$

4.6 Weight—the Force of Gravity; and the Normal Force

Weight is the gravitational force acting on an object, while the normal force is the support force exerted by surfaces.

• Weight: The force due to gravity, near Earth’s surface, is given by:

$W = m g$

• Normal Force: The force exerted perpendicular to a surface, balancing the weight when at rest.

• Equilibrium: For an object at rest, the net force is zero; normal force balances gravity.

4.7 Solving Problems with Newton’s Laws: Free-Body Diagrams

Free-body diagrams are essential tools for visualizing and solving problems involving forces.

• Steps:

  1. Draw a sketch of the situation.

  2. Draw a free-body diagram for each object, showing all forces with accurate magnitudes and directions.

  3. Label each force.

  4. Resolve vectors into components.

  5. Apply Newton’s second law to each component.

  6. Solve for unknowns.

• Tension: When a cord or rope pulls on an object, the force is called tension.

4.8 Problems Involving Friction and Inclines

Friction is a force that opposes motion between surfaces. It is characterized by coefficients that depend on the materials involved.

• Kinetic Friction: The force opposing motion when surfaces slide past each other.

• Equation:

$f_k = \mu_k N$

• Static Friction: The force preventing motion between stationary surfaces.

• Equation:

$f_s \leq \mu_s N$

• Coefficients of Friction: $\mu_k$ (kinetic) and $\mu_s$ (static) depend on the pair of surfaces.

• Table: Coefficients of Friction

Surface Pair	Static ($\mu_s$)	Kinetic ($\mu_k$)
Steel on steel	0.74	0.57
Wood on wood	0.50	0.30
Glass on glass	0.94	0.40
Additional info:	Values depend on surface conditions	Values depend on surface conditions

• Inclined Planes: An object on an incline experiences three forces: normal force (perpendicular), friction (parallel), and gravity (downward).

• Static vs. Kinetic: Static friction increases with applied force up to a maximum; then kinetic friction takes over as the object moves.

Summary of Chapter 4

• Newton’s First Law: If the net force is zero, an object remains at rest or moves at constant velocity.

• Newton’s Second Law: $\Sigma F = m a$

• Newton’s Third Law: $F_{AB} = -F_{BA}$

• Weight: $W = m g$

• Friction: $f_k = \mu_k N$ (kinetic), $f_s \leq \mu_s N$ (static)

Example: A block sliding down an incline is subject to gravity, normal force, and friction. If at rest, static friction balances the other forces; if moving, kinetic friction applies.