Dynamics I: Motion Along a Line – Comprehensive Study Notes

Dynamics I: Motion Along a Line

Concept of Equilibrium

Equilibrium in physics refers to the state in which all forces acting on an object cancel each other out, resulting in no acceleration. This does not necessarily mean the object is stationary; it may be moving at constant velocity.

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

• 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

  2. Write $\Sigma F = ma$

  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 $\Sigma F = ma$ to solve

• Example: Calculating normal force when pushing or pulling a book vertically.

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 components.

• Steps:

  1. Draw FBD

  2. Decompose forces into x and y components

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

  4. Solve for unknowns

• Example: A box suspended by two cables; calculate tension in each cable.

2D Forces in the Horizontal Plane

When forces act only in the horizontal plane, vertical forces (weight, normal) are not relevant for horizontal motion. Forces at angles must be decomposed.

• Key Point: $N = 0$ for purely horizontal forces.

• Example: Calculating net force and acceleration for a block pulled by two forces at different angles.

Weight Force and Gravitational Acceleration

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

• Weight: $W = mg$

• Gravitational Acceleration: $g$ varies by location (e.g., $g_{Earth} = 9.8\,\text{m/s}^2$, $g_{Moon} = 1.62\,\text{m/s}^2$)

• 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

Vertical Forces and Acceleration in the Y-Axis

Vertical forces can cause objects to accelerate along the y-axis. The net force determines the direction and magnitude of acceleration.

• Key Point: $\Sigma F_y = ma_y$

• Example: Calculating acceleration for a block pulled vertically by a string with different tension values.

Kinetic Friction

Kinetic friction is a resisting force that occurs when two surfaces slide against each other. It opposes the direction of motion.

• 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 box moving on a surface

Static Friction

Static friction prevents an object from starting to move. It acts in the direction opposite to the applied force until a threshold is reached.

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

• Comparison: $\mu_s \geq \mu_k$; harder to start moving than to keep moving

• 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, use both force equations and kinematic equations.

• Force Equation: $\Sigma F = ma$

• 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 applied force for a block accelerating on a frictionless surface

Summary Table: Friction Types and Properties

Type	Formula	Direction	Threshold
Kinetic Friction	$f_k = \mu_k N$	Opposite to motion	None
Static Friction	$f_{s,\text{max}} = \mu_s N$	Opposite to applied force	Maximum value before motion starts

Additional info: These notes expand on brief points with academic context, definitions, and examples to ensure completeness and clarity for exam preparation.