Fundamental Mechanics Concepts and Problem-Solving for PHYS 101

Mechanics Fundamentals and Problem-Solving

Newton's Laws of Motion

Newton's Laws form the foundation of classical mechanics, describing the relationship between forces and motion.

• First Law (Law of Inertia): An object remains at rest or in uniform motion unless acted upon by a net external force.

• Second Law: The net force on an object is equal to the mass of the object multiplied by its acceleration. $\sum \vec{F} = m\vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

• Gravitational Force: The force between two masses is given by $F_g = G \frac{M_1 M_2}{r^2}$ where $G = 6.67 \times 10^{-11} \text{N} \cdot \text{m}^2 / \text{kg}^2$.

Free-Body Diagrams

Free-body diagrams are graphical representations used to visualize the forces acting on an object.

• Key Forces: Tension ($T$), Normal force ($N$), Weight ($mg$), Friction ($f$), Applied force ($F_{applied}$).

• Application: Used to analyze problems involving equilibrium, acceleration, and rotational motion.

• Example: In drag racing, free-body diagrams help determine the forces on a tire during a burnout and during acceleration.

Kinematics Equations

Kinematics describes the motion of objects without considering the forces that cause the motion.

• Average Velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Average Acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$

• Equations of Motion (constant acceleration):

  • $v_f = v_i + a t$

  • $x_f = x_i + v_i t + \frac{1}{2} a t^2$

  • $v_f^2 = v_i^2 + 2 a (x_f - x_i)$

• Vectors:

  • Magnitude: $|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

  • Direction: $\tan \theta = \frac{A_y}{A_x}$

Rotational Motion

Rotational motion involves objects rotating about an axis, described by angular quantities.

• Angular Velocity: $\omega = \omega_i + \alpha t$

• Angular Displacement: $\theta = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$

• Rotational Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$

• Torque: $\tau = r F_{\perp}$

Apparent Weight in Accelerating Systems

Apparent weight is the normal force experienced by a person in an accelerating system, such as an elevator.

• Formula: $N = m(g + a)$, where $a$ is the acceleration of the system.

• Example: If an elevator accelerates upward, apparent weight increases; if downward, it decreases.

Force and Motion Graphs

Graphs of velocity vs. time and acceleration vs. time are used to interpret and predict motion.

• Constant Velocity: Horizontal line on $v$ vs. $t$ graph.

• Constant Acceleration: Sloped line on $v$ vs. $t$ graph.

• Changing Acceleration: Curved or segmented lines.

• Example: Matching car motion descriptions to velocity-time graphs.

Friction and Coefficients

Friction is a resistive force that opposes motion between surfaces.

• Static Friction: $f_s \leq \mu_s N$

• Kinetic Friction: $f_k = \mu_k N$

• Coefficients: Depend on material pairs; see table below.

Material	Static ($\mu_s$)	Kinetic ($\mu_k$)
Rubber on concrete	1.0	0.8
Steel on steel	0.74	0.57
Ice on ice	0.1	0.03
Wood on wood	0.5	0.3
Glass on glass	0.9	0.4
Additional info: Table inferred from exam sheet.

Problem-Solving Applications

Physics problems often require applying concepts to real-world scenarios.

• Elevator Problems: Use kinematics and Newton's laws to determine apparent weight and acceleration.

• Pulley Systems: Apply force analysis and kinematics to find unknown masses or accelerations.

• Circular Motion: For objects moving in a circle, centripetal force is required: $F_c = m \frac{v^2}{r}$

• Tension in Wires: Analyze forces in multiple directions for objects in equilibrium or circular motion.

Examples and Applications

• Moon's Orbit: The Moon orbits Earth due to gravitational attraction, not because of Earth's gravity alone or other planets.

• Drag Racing: Free-body diagrams help analyze tire forces during burnouts and acceleration.

• Reaction Forces: When pushing a crate, the reaction force is equal and opposite to the applied force, per Newton's third law.

• Net Force Calculation: For a ball moving horizontally, net force is calculated using $F = m a$.

Summary Table: Key Equations

Concept	Equation (LaTeX)
Average Velocity	$v_{avg} = \frac{\Delta x}{\Delta t}$
Average Acceleration	$a_{avg} = \frac{\Delta v}{\Delta t}$
Kinematics	$x_f = x_i + v_i t + \frac{1}{2} a t^2$
Newton's Second Law	$\sum \vec{F} = m\vec{a}$
Friction	$f_k = \mu_k N$
Centripetal Force	$F_c = m \frac{v^2}{r}$
Torque	$\tau = r F_{\perp}$
Gravitational Force	$F_g = G \frac{M_1 M_2}{r^2}$

Additional info: These notes expand on the exam questions and equation sheet, providing context and explanations for each major topic covered in PHYS 101 mechanics.