Mechanics: Work & Energy, Impulses & Collisions, Statics of Rigid Bodies

1. Work & Energy

Think of Energy as the "currency" of the universe and Work as the "transaction."

Basic: Definitions

Work (W): Occurs when a force causes displacement.

$$W=Fs\cos \left(\theta \right)$$

Intermediate: The Work-Energy Theorem

If you do work on an object, you change its kinetic energy. This is a scalar shortcut to avoid messy vector calculus.

$$W_{\mathrm{net}}=\Delta K=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$$

Expert: Conservative Systems & Power

In a closed system with no friction, the sum of Potential (U) and Kinetic (K) energy is constant. Power is simply how fast you spend that energy:

$$P=\frac{dW}{dt}=\mathbf{F}\cdot \mathbf{v}$$

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2. Impulses & Collisions

While Work happens over distance, Impulse happens over time.

Basic: Momentum (p)

$$p=mv$$

Intermediate: The Impulse-Momentum Theorem

Impulse (J) is the change in momentum. In a collision, even if kinetic energy is lost to sound or heat, momentum is always conserved in an isolated system.

$$J=\int Fdt=\Delta p$$

Expert: Collision Coefficient

The "bounciness" of a collision is defined by the Coefficient of Restitution (e):

$$e=\frac{v_{2f}-v_{1f}}{v_{1i}-v_{2i}}$$

Collision Type	Momentum	Kinetic Energy	e Value
Perfectly Elastic	Conserved	Conserved	e = 1
Inelastic	Conserved	Lost	0 < e < 1
Perfectly Inelastic	Conserved	Max Loss	e = 0

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3. Statics of Rigid Bodies

Statics is the art of making sure things don't move. For a rigid body (one that doesn't bend), we analyze two types of equilibrium.

The Two Conditions

• Translational: The sum of all forces is zero.
  $\sum F_x=0,\sum F_y=0$
• Rotational: The sum of all moments (torques) is zero.
  $\sum \tau =0$

Expert Tip: When solving statics problems, always pick a pivot point where the most unknown forces act. This cancels them out of your torque equation, leaving you with fewer variables!

Moments & Couples

A Moment is force times the perpendicular distance from the pivot. A Couple consists of two parallel forces of equal magnitude but opposite direction, creating pure rotation without translation.

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Final Summary Table

Scenario	Primary Tool
Objects moving through distance	Work-Energy
Rapid impacts or explosions	Impulse-Momentum
Structures at rest	Statics / Equilibrium