A.2.1 Newton's Laws of Motion

Newton’s Laws of Motion

1. You’re riding a skateboard.
2. As you push off the ground, you accelerate forward.
3. But what if you stop pushing? You gradually slow down and eventually come to a halt. Why does this happen?

This simple scenario illustrates the core principles of Newton’s laws of motion, which describe how objects move and interact with forces.

First Law of Motion: Inertia

Understanding Inertia and Its Implications

This principle is known as inertia.

1. Objects at Rest: A book on a table remains stationary unless a force (like a push) moves it.
2. Objects in Motion: A rolling ball continues to move in a straight line at constant speed unless friction or another force slows it down.

Second Law of Motion: $F=ma$

Deriving and Applying $F=ma$ to Linear Motion Scenarios

Newton’s second law of motion provides a quantitative description of how forces affect motion:

Mathematically, this is expressed as:

$${\overrightarrow{F}}_{\text{net}}=m\overrightarrow{a}$$

where:

• $F$ is the net force acting on the object (in newtons, N).
• $m$ is the mass of the object (in kilograms, kg).
• $a$ is the acceleration of the object (in meters per second squared, m/s²).

Why $F=ma$ Matters

1. Predicting Motion: By knowing the forces acting on an object, you can predict how it will move.
2. Designing Systems: Engineers use $F=ma$ to design everything from cars to rockets, ensuring they perform as expected under various forces.

Third Law of Motion: Action-Reaction Pairs

Analyzing Examples Such as Propulsion and Collisions

This means that forces always occur in pairs, known as action-reaction pairs.

• Propulsion:
  • When a rocket expels gas backward, the gas pushes the rocket forward with an equal and opposite force.
  • This is how rockets move in space.
• Collisions:
  • When a car crashes into a wall, the car exerts a force on the wall, and the wall exerts an equal and opposite force on the car.
  • This is why both the car and the wall experience damage.

Contact Forces and Field Forces

Contact Forces

• Normal Force ($F_N$):
  • The normal force acts perpendicular to a surface, supporting an object resting on it.
  • For objects on a horizontal surface, $F_N=mg$, where $m$ is mass and $g$ is gravitational acceleration.
  • It prevents objects from penetrating the surface.
• Frictional Force ($F_f$):
  • Friction opposes motion between surfaces in contact.

• Static Friction:
  • Prevents the initiation of motion, with a maximum value of $F_f\le {\mu }_s{F}_N$.
• Kinetic Friction:
  • Resists sliding motion, given by $F_f={\mu }_k{F}_N$.
  • Friction is essential for walking, driving, and holding objects.
• Tension:
  • Tension transmits force through ropes or cables, acting along their length.
  • It appears in systems like pulleys and hanging objects.
• Elastic Restoring Force ($F_e$):
  • Defined by Hooke’s Law, $F=-kx$, where $k$ is the spring constant and $x$ is displacement.
  • This force resists deformation and returns elastic objects to equilibrium.
• Viscous Drag Force ($F_d$):
  • A resistive force exerted by fluids, calculated as $F_d=6\pi \eta rv$, where $\eta$ is the fluid's viscosity, $r$ is the object's radius, and $v$ is velocity.
  • It is crucial in fluid dynamics and terminal velocity calculations.

• Buoyant Force ($F_b$):
  • The upward force exerted by a fluid on a submerged object, described by Archimedes’ principle: $F_b=\rho Vg$, where $\rho$ is the fluid density, $V$ is the displaced volume, and $g$ is gravitational acceleration.

Field Forces

• Gravitational Force ($F_g$):
  • The attractive force between two masses, calculated near Earth’s surface as $F_g=mg$.
  • It governs free fall, planetary motion, and tides.
• Electric Force ($F_e$):
  • The force between charges, given by Coulomb’s law: $$F_e=k\frac{|q_1{q}_2|}{r^2}$$ where $k$ is Coulomb's constant, $q_1$ and $q_2$ are charges, and $r$ is their separation.
  • It drives static electricity, circuits, and atomic interactions.
• Magnetic Force ($F_m$):
  • The force on a moving charge in a magnetic field: $$F_m=qvB\sin \theta$$ where $q$ is charge, $v$ is velocity, $B$ is field strength, and $\theta$ is the angle between them.
  • It explains phenomena like current-carrying wires, electric motors, and particle motion in magnetic fields.

Free-Body Diagrams

Visualizing and Resolving Forces Acting on a Body

• Free-body diagrams are essential tools for visualizing the forces acting on an object.
• They help you analyze both equilibrium and non-equilibrium situations.
  1. Identify the Object: Focus on the object of interest and isolate it from its surroundings.
  2. Draw the Forces: Represent each force acting on the object as an arrow.
  3. Label the Forces: Clearly label each force, such as gravitational force ($F_g$), normal force ($F_N$), frictional force ($F_f$), and applied force ($F_a$).

Equilibrium vs. Non-Equilibrium States

• Equilibrium:
  • The net force acting on the object is zero, resulting in no acceleration.
  • The object remains at rest or moves at a constant velocity.
• Non-Equilibrium:
  • The net force is not zero, causing the object to accelerate in the direction of the net force.

Reflection and Broader Implications

Newton’s laws of motion provide a foundational framework for understanding the physical world. They explain everything from the motion of everyday objects to the dynamics of celestial bodies.