PSAD Formulas 01

Engineering Mechanics

Resultant of a Force System: Components

R_x = ΣFcosα

R_y = ΣFsinα

α is angle wrt x-axis

Resultant of a Force System: Magnitude and Direction

R = √(R_x² + R_y²)

Θ = tan^-1(R_y / R_x)

α is angle wrt x-axis

Moment about a Point

M = Fd

Centroid and Effective Forces

Concentrated Load (Point): F = P

Uniformly Distributed Load: F = wL @ L/2

Uniformly Varying Load (Triangular): F = wL/2 @ L/3

Squared Property of Parabola (Parabolic Cable)

d₁² / h₁ = d₂² / h₂

d = distance from lowest point to support

h = height of supports from lowest point

Parabolic Cable: Minimum and Maximum Tension

T_min = (w d_max² / 2 h_max)

T_max = √((T_min)² + (w d_max)²)

Length of Catenary Cable

S = (T₀ / W₀) sinh(W₀ x / T₀)

Catenary Cable: Tension at any Point

T = T₀ cosh(W₀ x / T₀)

Catenary Cable: Vertical Distance of Lowest Point to any Point of the Cable

y = (T₀ / W₀) [cosh(W₀ x / T₀) - 1]

Friction Force

F_f = µ_s N (static friction)

F_f = µ_k N (dynamic friction)

Angle of Friction

tanΘ = F / N

when Θ = α, tanα = µ = F / N

Belt Friction

T₂ / T₁ = e^µß

T₂ > T₁

ß = subtended angle in radians

Rectangle: Area, Centroid, Moment of Inertia

A = bh

Centroid = (b/2, h/2)

I_x = bh³ / 12

I_y = hb³ / 12

Triangle: Area, Centroid, Moment of Inertia

A = bh / 2

Centroid = (b/3, h/3)

I_x = bh³ / 36

I_y = hb³ / 36

Circle: Area, Centroid, Moment of Inertia

A = πr² = πd² / 4

Centroid = (d/2, d/2)

I_x = πd⁴ / 64

I_y = πd⁴ / 64

Quarter Circle / Spandrel

A = πr² / 4

Centroid = (4r/3π, 4r/3π)

I_x = 0.055r⁴

I_y = 0.055r⁴

Rectilinear Motion: Uniform Motion

s = vt

a = 0

Rectilinear Motion: Uniformly Accelerated Motion

a = constant

v = v₀ ± at

s = v₀ t ± at² / 2

v² = v₀² ± 2a (x - x₀)

Rectilinear Motion: Instantaneous vs. Average

Instantaneous Velocity: v = dx / dt

Instantaneous Acceleration: a = dv / dt = d²x / dt²

Average Velocity: v_ave = (x - x₀) / (t - t₀)

Average Acceleration: a_ave = (v - v₀) / (t - t₀)

Rectilinear Motion: Free Fall

v₂ = v₁ ± gt

v₂² = v₁² ± 2gy

y = v₁t ± gt² / 2

t = √(2y / g)

y_max = v₁² / 2g

(+) - upwards

(-) - downwards

Projectile Motion

v_0x = v₀ cosΘ

v_0y = v₀ sinΘ

x = v_0x t

t = v₀ sinΘ / g + √(2h_max / g)

h_max = y_max + h

y_max = v₀²sin²Θ / 2g

Projectile Motion: General Formula for height of projectile

y = xtanΘ - gx² / (2v₀²cos²Θ)

Rotational Motion

ω = dΘ / dt

α = dω / dt

s = rΘ

v = rω

Rotational Motion: Acceleration Components

a_T = αr

a_N = ω²r = v² / r

a = √(a_T² + a_N²)

Work

W = F d

Power

P = W / t

Gravitational Potential Energy

PE = m g h

Spring Potential Energy

PE = kx² / 2

Kinetic Energy

KE = mv² / 2

Conservation of Mechanical Energy

PE₁ + KE₁ = PE₂ + KE₂

For PE = KE: m g h = mv² / 2

v = √2gh

Linear Momentum and Impulse

p = m × v

Conservation: mv₁ = mv₂

J = F × t

Coefficient of Resitution

e = - (v_B2 - v_A2) / (v_B1 - v_A1)

Particle Kinetics: Centripetal Force

F_cf = mv² / r

Particle Kinetics: Banked Curves

Friction is Considered: tan (Θ + Φ) = v² / gR

Car is Slipping: tan (Θ - Φ) = v² / gR

Design Angle of Banking: tan (Θ) = v² / gR