Study guide: Fundamentals of Mechanics

Mechanics Revision Sheet

1. 📌 Essentials

• Newton's second law: $= ma$ — fundamental for dynamics.
• Equilibrium: $\sum F = 0$, $\sum \tau = 0$ — static conditions.
• Work-energy theorem: net work equals change in kinetic energy.
• Impulse: $J = F \Delta t = \Delta p$ — describes collisions.
• Moment of inertia ($I$): depends on mass distribution; key in rotational motion.
• Simple harmonic motion: $x(t) = A \cos (\omega t + \phi)$, $T = 2\pi \sqrt{\frac{m}{k}}$.
• Conservation of energy: total mechanical energy remains constant in ideal systems.
• Friction: $F_f = \mu N$; static vs kinetic.
• Power: $P = \frac{W}{t}$ — rate of work.
• Mechanical advantage: ratio of load to effort in levers and pulleys.

2. 🧩 Key Structures & Components

• Displacement / Velocity / Acceleration — describe motion in kinematics.
• Forces — gravity, friction, tension, normal force.
• Mass and Shape — influence moment of inertia ($I$).
• Springs — characterized by spring constant $k$, oscillate in SHM.
• Levers / Pulleys — mechanical systems to amplify force.
• Friction surfaces — static and kinetic coefficients ($\mu_s, \mu_k$).
• Rotational Axes — points about which objects rotate.
• Energy reservoirs — kinetic energy, potential energy (gravitational).

3. 🔬 Functions, Mechanisms & Relationships

• Linear to rotational conversion: torque ($\tau$) causes angular acceleration ($\alpha$), related by $\tau = I \alpha$.
• Force application: forces produce acceleration ($a$), which can be translated into angular acceleration via torque.
• Energy transfer: work done by forces changes kinetic or potential energy.
• Impulse transfer: force applied over time changes momentum ($p = mv$).
• Equilibrium: sum of forces and moments equals zero, maintaining static or constant motion.
• Oscillations: restoring force proportional to displacement ($F = -kx$) causes SHM.
• Hierarchy: forces act on particles, which are part of rigid bodies, which rotate or translate.

4. 🗂️ Hierarchical Diagram (ASCII)

Mechanics
 ├─ Kinematics
 │    ├─ Displacement
 │    ├─ Velocity
 │    └─ Acceleration
 ├─ Dynamics
 │    ├─ Forces
 │    │    ├─ Gravity
 │    │    ├─ Friction
 │    │    └─ Tension
 │    └─ Newton's Laws
 ├─ Statics
 │    ├─ Equilibrium
 │    └─ Force & Moment Balance
 ├─ Work & Energy
 │    ├─ Work
 │    ├─ KE & PE
 │    └─ Conservation
 ├─ Impulse & Momentum
 │    └─ Change in momentum
 └─ Rotational Mechanics
      ├─ Torque
      ├─ Moment of inertia
      ├─ Angular velocity
      └─ Angular momentum

5. ⚠️ High-Yield Pitfalls & Confusions

• Confusing $F = ma$ with static friction limits.
• Mixing linear and rotational quantities (e.g., torque vs force).
• Assuming energy conservation in non-conservative systems.
• Overlooking the difference between static and kinetic friction.
• Misidentifying the axis of rotation when calculating $I$.
• Forgetting to include the $\sin \theta$ component in torque calculations.
• Assuming $I$ is the same for all shapes; it varies with geometry.
• Ignoring the directionality of forces and motion.
• Misapplying SHM formulas outside their valid context.
• Overlooking the effect of external torques or forces in equilibrium.

6. ✅ Final Exam Checklist

• Understand and apply $F = ma$ in linear motion.
• Know equilibrium conditions: $\sum F = 0$, $\sum \tau = 0$.
• Calculate work, kinetic energy, potential energy, and apply conservation.
• Use impulse-momentum theorem for collision problems.
• Determine moment of inertia ($I$) for common shapes.
• Compute torque: $\tau = rF \sin \theta$.
• Derive and analyze simple harmonic motion parameters.
• Differentiate between static and kinetic friction.
• Calculate power: $P = \frac{W}{t}$.
• Recognize mechanical advantage in levers and pulleys.
• Convert between linear and rotational quantities.
• Identify forces causing rotational acceleration.
• Solve problems involving energy transfer in oscillations.
• Draw free-body diagrams accurately.
• Apply the hierarchy of forces and moments.
• Use appropriate units and check for consistency.

End of Revision Sheet