3. Mechanics of fluids, gases and solid bodies

Mechanisms based on balance of forces

Mechanisms using force equilibrium, including levers, slopes, screws, and wedges.

Momentum of force relative to the axis of rotation

A vector that measures the rotational effect of a force

Rotational kinetic energy of a homogeneous body

Energy from an object's rotation, proportional to the square of angular velocity and moment of inertia

Rolling resistance force

Force opposing a rolling body's motion, proportional to compressive force and inversely to the rolling resistance shoulder; lower than sliding friction

Pressure exerted by external force on ideal fluid

Pressure transmitted uniformly in an ideal fluid when a force is applied, inversely proportional to the surface area

Internal flow in a liquid

Fluid movement within a liquid, dependent on temperature, dynamic viscosity, and density.

Frictional forces (sliding)

Forces opposing motion when a body slides, resulting from surface heterogeneity and acting opposite to motion

Rotational effects of forces acting on a rigid body

A rigid body is in equilibrium or has constant rotational motion when the net moment of force is zero.

Center of gravity

Point where total gravitational forces act on a body; geometric center for regular, homogeneous solids; can be outside the body.

Normal atmospheric pressure

Standard atmospheric pressure: 1013.25 hPa, 101.325 kPa, or 101325 Pa.

Buoyancy force

Upward force exerted by a fluid on an object; greatest in the densest fluid (e.g., water).

Equilibrium position (body stays after deflection)

Equilibrium where a body remains in any new position after deflection; neutral or indifferent equilibrium.

Ideal liquid flow (horizontal tube)

In ideal liquid flow in a horizontal tube, the total of kinetic and compressional energy is constant.

Rigid body

Ideal solid with unchangeable shape/volume under external conditions; non-zero shear stiffness

Hydrostatic pressure on bottom of well

Liquid pressure at rest, depending on liquid column height, not mass or surface area.

Archimedes' principle

The principle applies to liquid density and displaced volume..

Formula for Moment of Force (Torque) formula

T = F.d (where d is the perpendicular distance from the pivot to the line of action of the force

Torque measured in

Newton-meters (Nm)

Rotational energy measure in

Joules (J)

ω (omega) measured in

radians per second (rad/s)

Rotational Kinetic Energy formula

Ekrot = ½ Iω², where I= moment of inertia and w, angular velocity how fast an object is rotating

Rolling Resistance Force formula

Fr = (k/R) Fn (where k is the coefficient of rolling resistance, R is the radius of the rolling body, and Fn is the normal force or compressive force)

Rolling Force Measure in

Netwons (N)

Hydrostatic Pressure (Ph) formula

Ph = ρgh, where p= density of the fluid, g=acceleration due to gravity and h=height/depth of the fluid

Buoyant Force formula

Fb = ρfluidgVdisplaced,where pfluid=density of fluid, g=acceleration due to gravity and vdisplaced=volume of fluid dispalced

Pascal's Principle (Hydraulic Systems) formula

F₁/A₁ = F₂/A₂,

where,

• F1​: Force applied to the first (often smaller) piston (N).

• A1​: Cross-sectional area of the first piston (m²).

• F2​: Force exerted by the second (often larger) piston (N).

• A2​: Cross-sectional area of the second piston (m²).

Bernoulli's Principle (Horizontal Fluid Flow) formula

P + ½ ρv² = constant

Bernoulli's Principle (General Fluid Flow) formula

P + pgh + ½ pv² = constant

where,

• P: Static pressure of the fluid (Pa).

• p (rho): Density of the fluid (kg/m³).

• g: Acceleration due to gravity (m/s²).

• h: Height or depth of the fluid (vertical position) (m).

• v: Velocity (speed) of the fluid flow (m/s)

Torricelli's Law formula

v = √2gh

• v: Velocity of the flowing liquid (m/s).

• g: Acceleration due to gravity (m/s²).

• h: Height of the liquid surface above the hole (m).

Density formula

ρ = m/V

• ρ (rho): Density (kg/m³).

• m: Mass of the substance (kg).

• V: Volume occupied by the substance (m³

Weight (Force of Gravity) formula

W = mg where, m=mass of object in kg and g= acceleration due to gravity

Equation of Continuity formula

A₁v₁ = A₂v₂

• A1​: Cross-sectional area of the flow at point 1 (m²).

• v1​: Velocity of the fluid flow at point 1 (m/s).

• A2​: Cross-sectional area of the flow at point 2 (m²).

• v2​: Velocity of the fluid flow at point 2 (m/s).