Technical Math & Science Foundations for Mechanical Engineering

What is a unit system?

A consistent set of base units used to measure physical quantities.

What is SI?

The International System of Units, built around meters, kilograms, seconds, kelvin, and amperes.

What are US customary units?

A system of measurement that includes feet, pounds, and seconds, often used in certain industries.

What is dimensional analysis?

A method used to check whether an equation is physically plausible by comparing dimensions.

What is force in SI units?

1 N = 1 kg·m/s².

How is energy defined in relation to force?

Energy is defined as work done by a force over a distance (1 J = 1 N·m).

What does pressure measure in SI units?

Pressure is defined as force per unit area (1 Pa = 1 N/m²).

What is the principle of factor-label conversion?

A method of unit conversion where conversion factors are multiplied as fractions equal to 1.

How does significant figures relate to measurements?

Significant figures communicate measurement precision and should not imply more precision than the least precise input.

What are common derived SI units?

Units like newton (force), joule (energy), watt (power), and pascal (pressure).

What is engineering notation?

Scientific notation where exponents are multiples of 3 to match SI prefixes.

What is Young’s modulus?

A material property describing stiffness, defined as the ratio of stress to strain.

What is a scalar?

A quantity that has only magnitude, such as mass or temperature.

What is a vector?

A quantity that has both magnitude and direction, such as force or velocity.

What does Hooke's Law state?

Stress is proportional to strain within the elastic limit of a material.

What is the formula for axial stress?

Axial stress (σ) is defined as σ = F/A, where F is the force applied and A is the cross-sectional area.

What are the equilibrium conditions for a particle?

For a particle in equilibrium, the net force must satisfy ∑Fx = 0 and ∑Fy = 0.

What does the term 'equilibrium' mean in a mechanical context?

A state where the sum of forces and moments acting on a body is zero.

What is Bernoulli's equation?

An equation relating pressure, velocity, and height in fluid dynamics, expressed as P + 1/2ρv² + ρgh = constant.

What does kinetic energy measure?

The energy that an object possesses due to its motion, given by KE = 1/2 mv².

What is potential energy in gravitational contexts?

The energy stored in an object due to its height above a reference point, given by PE = mgh.

What is the significance of a safety factor?

A design measure that provides a margin of safety for allowable loads and stresses.

How does fluid pressure change with depth?

Hydrostatic pressure increases with depth according to ΔP = ρgΔh.

What is the formula for calculating power?

Power (P) is calculated as P = W/t, where W is work done and t is time.

What is the continuity equation for fluid flow?

A principle stating that for incompressible flow, the mass flow rate must remain constant, expressed as Q = Av.

What factors compose a free-body diagram?

A diagram indicating all external forces and moments acting on an object.

What is a moment in mechanics?

A measure of the tendency of a force to cause rotation about a point, calculated as M = rFsin(θ).

What defines strain?

Strain (ε) is the relative change in length of a material, expressed as ε = ΔL/L₀.

What is the ideal gas law?

The relationship PV = nRT describes the behavior of ideal gases, where P is pressure, V is volume, n is moles, R is the gas constant, and T is temperature.

What is the main equation used to calculate work?

Work (W) is defined as W = Fd cos(θ), where F is the force applied, d is the distance moved, and θ is the angle between the force and direction of movement.

What is the role of significant figures in reporting results?

They indicate the precision of a measurement and should reflect the accuracy of the least precise input.

What does the term 'buoyancy' refer to?

The upward force exerted by a fluid on an immersed object, equal to the weight of the fluid displaced.

How does temperature relate to heat?

Temperature is a measure of thermal energy, while heat is energy transferred due to a temperature difference.

What is the relation of torque to force and distance?

Torque (M) is calculated as the product of force (F) and the perpendicular distance from the pivot point (r), expressed as M = rF.

What does it mean to say that energy is conserved?

Energy cannot be created or destroyed; it can only be transformed from one form to another.

What is the significance of dimensional consistency in equations?

Equations must remain dimensionally consistent, meaning both sides of the equation must have the same units.

What is the purpose of using a data graph?

Graphs visually represent relationships between variables, indicating trends and patterns in data.

What is the first law of thermodynamics?

Energy cannot be created or destroyed; it can only change forms, often referred to as the principle of energy conservation.

What is a common error when using dimensional analysis?

Assuming dimensional consistency without verifying that all derived units match the fundamental units in the equation.

What are significant figures?

The digits that carry meaning contributing to its precision, excluding leading or trailing zeros.

How do you define mechanical power?

The rate at which work is done or energy is transferred, typically measured in watts.

What are the forms of energy transfer in thermal processes?

Energy can be transferred as heat due to temperature differences or work due to mechanical processes.

What does a safety factor of 2 imply?

It means the allowable stress is half of the ultimate strength, providing a margin for unexpected conditions.