Newton's Laws of Motion

Newton's Laws of Motion

Newton's laws of motion are the foundation of classical mechanics, describing the relationship between the motion of an object and the forces acting on it.

Force and Mass

• Force: An interaction that causes an object to accelerate (a push or a pull). It is a vector quantity.
• Mass: A measure of an object's inertia, or resistance to changes in motion. It is a scalar quantity.

Newton's First Law (Law of Inertia)

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. $\sum \mathbf{F} = 0 \implies \mathbf{a} = 0$

Newton's Second Law

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. $\sum \mathbf{F} = m \mathbf{a}$ where $\sum \mathbf{F}$ is the vector sum of all forces.

Engineering Context: In structural engineering, we often deal with static equilibrium where $\mathbf{a} = 0$, so $\sum \mathbf{F} = 0$. This is the basis for analyzing trusses and beams. The National Structural Code of the Philippines (NSCP) 2015 defines various load combinations (Dead Loads, Live Loads, Earthquake Loads) that structures must resist to maintain this equilibrium.

Newton's Third Law

For every action, there is an equal and opposite reaction. $\mathbf{F}_{AB} = -\mathbf{F}_{BA}$ This means forces always occur in pairs.

Applications of Newton's Laws

To apply Newton's laws, we typically use Free Body Diagrams (FBD). An FBD shows all the forces acting on a single object.

Common Forces:

• Weight ($W$): The force of gravity. $W = mg$.
• Normal Force ($N$): The contact force exerted by a surface perpendicular to the object.
• Tension ($T$): The pulling force exerted by a string, rope, or cable.
• Friction ($f$): The force resisting relative motion between surfaces.

Friction and Drag

Friction

Friction is a force that opposes motion between two surfaces in contact.

• Static Friction ($f_s$): Prevents motion. $f_s \le \mu_s N$.
• Kinetic Friction ($f_k$): Acts during motion. $f_k = \mu_k N$.

where $\mu_s$ and $\mu_k$ are coefficients of static and kinetic friction, respectively.

Drag

Drag is the resistance force exerted by a fluid (liquid or gas) on an object moving through it. $D = \frac{1}{2} C \rho A v^2$ where:

• $C$ is the drag coefficient.
• $\rho$ is the fluid density.
• $A$ is the cross-sectional area.
• $v$ is the speed.

Drag is crucial in civil engineering for calculating wind loads on buildings and bridges (refer to NSCP 2015, Vol 1, Section 207 for Wind Loads).