Resistance = pL/A

Resistance is given by the equation R=pL/A, where p is resistivity, L is the length of the resistor, and A is the cross-sectional area of the resistor.

Dot Product

A • B = 

Cross Product

u X v =

Gravitational Force between two objects

[image]
where:

• F is the force between the masses;
• G is the gravitational constant (6.674×10⁻¹¹ N · (m/kg)²);
• m₁ is the first mass;
• m₂ is the second mass;
• r is the distance between the centers of the masses.

First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. Fnet = ma = 0

Second law: The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object: Fnet = ma.

Third law:When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The magnitude of the centripetal force on an object of mass m moving at tangential speed v along a path with radius of curvature r is:

[image]

where [image] is the centripetal acceleration. 

[image]               [image]

where

[image] is the torque vector and [image] is the magnitude of the torque,

r is the position vector (a vector from the origin of the coordinate system defined to the point where the force is applied)

F is the force vector,

× denotes the cross product,

θ is the angle between the force vector and the lever arm vector.

U = mgh

mass

gravitational constant

height

[image]

k = spring constant

Δx = magnitude of displacement from equilibrium

E = U + K

Total = potential + kinetic

Conservation of mechaincal energy

VS

Nonconservative work formula 

ΔE = ΔU + ΔK = 0

W_{nonconservative} = ΔE = ΔU + ΔK

W = F • d = Fdcos(θ)

W = Work

F = magnitude of the applied force

d = magnitude of the displacement through the force applied

θ = the angle between the applied force vector and the displacement vector

W = PΔV

Find the area under the curve with P on Y axis and V on the X.

Volume constant then no work.

Gas expands, W is positive

Gas compresses, W is negative

P = W/t = ΔE/t

Power equals work over time or the change in energy over time

Unit = J/sec

Mechanical advantage

If a and b are distances from the fulcrum to points A and B and if force F_A applied to A is the input force and F_B exerted at B is the output, the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by

[image]

Because the power flow is constant, the torque T_B and angular velocity ω_B of the output gear must satisfy the relation

[image]

which yields

[image]

Efficiency = W Out/ W In

=(load)(load distance)/(effort*effort distance)

The change in the linear dimension can be estimated to be:

[image]

This equation works well as long as the linear-expansion coefficient does not change much over the change in temperature [image], and the fractional change in length is small [image]. If either of these conditions does not hold, the equation must be integrated

If we already know the expansion coefficient, then we can calculate the change in volume

[image]

where [image] is the fractional change in volume (e.g., 0.002) and [image] is the change in temperature (50 °C).

In a non-cyclic process, the change in the internal energy of a system is equal to net energy added as heat to the system minus thenet work done by the system, both being measured in mechanical units. Taking ΔU as a change in internal energy, one writes

[image]

where Q denotes the net quantity of heat supplied to the system by its surroundings and W denotes the net work done by the system. This sign convention is implicit in Clausius' statement of the law given above. It originated with the study of heat engines that produce useful work by consumption of heat

ΔS = Q_rev/t

S is change in entrhopy

Q heat change in the process

T = temp

Mathematically, density is defined as mass divided by volume:^[1]

[image]

where ρ is the density, m is the mass, and V is the volume. 

Fg = density * Volume * gravity constant

For fluids

Specific gravity is the ratio of the density of a substance to the density of a reference substance; equivalently, it is the ratio of the mass of a substance to the mass of a reference substance for the same given volume.

It can be shown that true specific gravity can be computed from different properties:

[image]

The total pressure that is exerted on an object that is submerged in a fluid

Pressure = surface pressure + density*gravity acceleration*depth

P = P₀ +ρgz

P_gauge = P-P_atm = (p₀+ρgz)-p_atm

ρ  = density

g = acceleration due to gravity

z=depth

Seen in pistons

P = F1/A1 = F2/A2

F2=F1(A2/A1)

Archimedes Principle

Buoyancy

F_buoy = ρ_fluid*V_{fluid displace}*g

= ρ_fluid*Vsubmerged*g

Poiseuille equation

Flow through a pipe

[image]

where:

[image] is the pressure loss

[image] is the length of pipe

[image] is the dynamic viscosity

[image] is the volumetric flow rate

[image] is the radius

[image] is the velocity

Critical Velocity

 [image] 

Nr is a constant

η = viscosity

density

diameter

Coulomb's law

In its scalar form the law is:

[image] ,

where k_e is Coulomb's constant (k_e = 8.99×10⁹ N m² C⁻²), q₁ and q₂are the signed magnitudes of the charges, the scalar r is the distance between the charges.

E = F_e/q = kQ/r²

[image],

where [image] is Coulomb's constant, r is the distance between the point charges q & Q, and q & Q are the signed values of the charges

Electrical Potential

From Potential Energy

vs

From Source Energy

V = U/q

vs

V= kQ/r

Voltage

ΔV = 

Magnetic field from a 

Straight Wire

vs

Loop of wire

B = (μ₀I)/(2πr)

vs

B = (μ₀I)/(2r)

Magnetic Force on a...

Moving point charge

vs

Current-carrying wire

F_b = qvB*sin(θ)

vs

F_b = ILB*sin(θ)