Dynamics I: Motion Along a Line – Comprehensive Study Notes

Dynamics I: Motion Along a Line

Concept of Equilibrium

Equilibrium occurs when all the forces acting on an object balance each other, resulting in no acceleration. This does not necessarily mean the object is stationary; it may be moving at constant velocity.

• Equilibrium Condition: $\sum F = 0 \Leftrightarrow a = 0$

• Key Point: An object in equilibrium has zero acceleration, not necessarily zero velocity.

• Example: A box pulled by two equal forces at constant speed has zero acceleration.

Free-Body Diagrams (FBD) and Force Analysis

To analyze forces, draw a Free-Body Diagram (FBD) showing all forces acting on the object. Typical forces include weight (W), applied force (FA), tension (T), normal force (N), and friction (f).

• Steps:

  1. Draw FBD: W, FA, T, N, f

  2. Write $\sum F = ma$

  3. Solve for unknowns

• Example: A 2 kg book at rest on a table: Forces are weight and normal force.

Normal Force

The normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface.

• Key Properties:

  • Always perpendicular to the surface

  • No direct equation; must use $\sum F = ma$ to solve for N

• Example: A book pushed down or pulled up changes the normal force accordingly.

Equilibrium in Two Dimensions (2D)

In 2D equilibrium, all forces must cancel out in both the x and y axes. Forces acting at angles must be decomposed into their components.

• Steps:

  1. Draw FBD

  2. Decompose forces into x and y components

  3. Write $\sum F_x = ma_x$ and $\sum F_y = ma_y$

  4. Solve for unknowns

• Example: A box suspended by two cables requires solving for tension in each cable.

Weight Force and Gravitational Acceleration

Gravity acts on all objects near Earth, producing a force called weight and an acceleration called gravitational acceleration.

• Weight: $W = mg$

• Gravitational Acceleration: $g_{Earth} \approx 9.8\ \text{m/s}^2$, $g_{Moon} \approx 1.62\ \text{m/s}^2$

• Mass: Quantity of matter, does not change with location

• Weight: Force due to gravity, changes with location

• Example: Calculating weight on Earth and Moon for a given mass

Kinetic Friction

Kinetic friction is a resistive force that opposes the motion of two surfaces sliding against each other. It acts opposite to the direction of velocity.

• Formula: $f_k = \mu_k N$

• Coefficient of Kinetic Friction ($\mu_k$): Unitless, measures surface roughness (range: 0 to 1)

• Example: Calculating friction force and acceleration for a moving box

Static Friction

Static friction prevents an object from starting to move. It acts up to a maximum value, after which the object begins to move and kinetic friction takes over.

• Maximum Static Friction: $f_{s,\text{max}} = \mu_s N$

• Comparison: $\mu_s \geq \mu_k$ (static friction is always greater than or equal to kinetic friction)

• Example: Determining if a block will move based on applied force and static friction threshold

Solving 1D Motion Problems with Forces

Forces cause acceleration, which changes an object's speed or direction. To solve problems involving forces and motion, use Newton's Second Law and kinematic equations.

• Newton's Second Law: $\sum F = ma$

• Kinematic Equations (UAM):

  • $v_x = v_{0x} + a_x t$

  • $v_x^2 = v_{0x}^2 + 2 a_x \Delta x$

  • $\Delta x = v_{0x} t + \frac{1}{2} a_x t^2$

  • $\Delta x = \frac{1}{2}(v_{0x} + v_x)t$

• Example: Calculating force required to accelerate a block, or distance traveled by a car before stopping

Summary Table: Types of Friction

Type	Formula	When Active	Direction
Kinetic Friction	$f_k = \mu_k N$	Object is moving	Opposite to velocity
Static Friction	$f_{s,\text{max}} = \mu_s N$	Object is at rest	Opposite to potential motion

Summary Table: Force Variables vs. Motion Variables

Force Variables	Motion Variables
m, $\vec{F}$, $\vec{a}$	$\vec{v}$, $\Delta x$, $\Delta t$

Additional info: Academic context and expanded explanations were added to clarify concepts and ensure completeness for exam preparation.