Chapter 6: Dynamics I – Motion Along a Line (Study Notes)

Chapter 6: Dynamics I – Motion Along a Line

Section 6.1: Static and Dynamic Equilibrium

Equilibrium in physics refers to the state in which the sum of all forces acting on an object is zero. This section distinguishes between static equilibrium (no motion) and dynamic equilibrium (constant velocity).

• Static Equilibrium: An object is at rest and remains at rest. The net force is zero.

• Dynamic Equilibrium: An object moves with constant velocity. The net force is also zero.

• Condition for Equilibrium: $\sum \vec{F} = 0$

• Example: A book resting on a table is in static equilibrium; a car moving at constant speed on a straight road is in dynamic equilibrium.

Section 6.2: Using Newton's Second Law

Newton's Second Law relates the net force acting on an object to its acceleration. It is fundamental for analyzing motion along a line.

• Newton's Second Law: $\vec{F}_{\text{net}} = m \vec{a}$

• Application: To solve for acceleration, mass, or net force, identify all forces acting on the object and sum them vectorially.

• Example: If a 2 kg object experiences a net force of 10 N, its acceleration is $a = \frac{F_{\text{net}}}{m} = \frac{10}{2} = 5\ \text{m/s}^2$.

Section 6.3: Mass, Weight, and Gravity

Mass is a measure of an object's inertia, while weight is the force exerted by gravity on that mass. Gravity is the attractive force between masses.

• Mass (m): Scalar quantity representing the amount of matter in an object (kg).

• Weight (W): Force due to gravity, $W = mg$ where $g \approx 9.8\ \text{m/s}^2$.

• Gravity: The acceleration due to Earth's gravitational field.

• Example: A 5 kg object has a weight of $W = 5 \times 9.8 = 49\ \text{N}$.

Section 6.4: Static and Kinetic Friction

Friction is a force that opposes motion between two surfaces. Static friction prevents motion, while kinetic friction acts during motion.

• Static Friction ($f_s$): Maximum force before motion starts, $f_s \leq \mu_s N$.

• Kinetic Friction ($f_k$): Force during motion, $f_k = \mu_k N$.

• Coefficients: $\mu_s$ (static), $\mu_k$ (kinetic) are dimensionless and depend on materials.

• Normal Force (N): Perpendicular contact force.

• Example: If $\mu_s = 0.4$, $N = 100\ \text{N}$, then $f_s \leq 40\ \text{N}$.

Section 6.5: Drag Force and Terminal Speed

Drag force opposes the motion of objects through fluids (air or water). Terminal speed is reached when the drag force equals the force of gravity, resulting in zero net acceleration.

• Drag Force ($F_{\text{drag}}$): Often modeled as $F_{\text{drag}} = bv$ (linear) or $F_{\text{drag}} = cv^2$ (quadratic).

• Terminal Speed ($v_t$): The constant speed when $F_{\text{drag}} = mg$.

• Equation for Terminal Speed: $mg = cv_t^2 \Rightarrow v_t = \sqrt{\frac{mg}{c}}$

• Example: A falling skydiver eventually reaches terminal speed when air resistance balances weight.