Understanding Forces: Normal, Tension, Friction, and Rotation

Posted on Sep 23, 2025 in Naval Architecture and Marine Systems Engineering

Normal Force

N = -p

F = m * a

Inclined Surface

a = g * sin(θ)

Weight on the inclined plane:

Px = P * sin(θ)

Py = P * cos(θ)

Tension

T = -p

Two Strings Attached

With an angle:

T = p / (2 * sin(θ))

Two Strings Attached

Without an angle:

T = p / 2

Dragging an Object

T = F – m₁ * a

Pendulum in Wagon

x coord: T * sin(θ) = m * a

y coord: T * cos(θ) – m * g = 0

tan(θ) = a / g

T = m * g / cos(θ)

Elastic Force

F = K * Δx (stretch)

F = -K * Δx (compression)

Spring (Vertical)

F = p -> p = K * Δy

K = Δy / g

Friction Forces

Static Friction Force: F_s = μ_s * N

μ_s = tan(θ) -> μ = P * sin(θ) / P * cos(θ)

Kinetic Friction Force: F_k = μ_k * N

μ_k = (g * sin(θ) – a) / (g * cos(θ))

μ_k < μ_s

F – μ * m * g = m * a

a = (F – μ * m * g) / m

Case Studies

Two Bodies on a Table with Pulley (No Angle)

T – F_f = m₁ * a

P₂ – T = m₂ * a

Atwood Machine

a = ΔF / m_total = (m₂ – m₁) / (m₁ + m₂) * g

T – p = m * a

Object Moving Up an Inclined Plane

(Force acting on m)

F – Px – F_f = m * a

F – m * g * sin(θ) – μ_s * m * g * cos(θ) = m * a

Object Moving Down an Inclined Plane (Two Bodies, No Friction)

m₁ * g * sin(θ) – T = m₁ * a

m₂ * g – T = m₂ * a

Friction on Inclined Plane

x coord: g * sin(θ) – F_f = a

y coord: N – m * g * cos(θ) = 0

F_f = μ * m * g * cos(θ)

a = g * (sin(θ) – μ_k * cos(θ))

Elevator

Going Up: N – m * g = m * a

T – m * g = m * a

Going Down: N + m * g = m * a

T + m * g = m * a

Rotating Body Tied to a Rope (No Friction)

Body in a Rotating Horizontal Surface (No Friction)

ΣF = m * a

T + N + p = m * a

Components:

x: T = m * a

y: N – p = 0

T = m * v² / R

T = m * ω² * R

Body Tied to a Table with Another Below (No Angle)

ω = √(m₁ * g / (m₂ * R)) (m₁ = below table)

Body in a Rotating Vertical Plane

ΣF = m * a

T – p = m * a

T + p = m * a

p = m * v² / R

v = √(g * R)

Minimum velocity at the highest point:

v = √(m * g / (m * R))

a_n = v² / R

a_t = Δv / Δt

Body Rotating in a Horizontal Plane

ΣF = m * a

p + T = m * a

Components:

Tx = T * sin(θ)

Tx = m * a

T = m * (v² / R) / sin(θ)

Ty = T * cos(θ)

T – p = 0

T = m * g / cos(θ)

tan(θ) = v² / (R * g)

sin(θ) = R / l

If you just give angle, radius and tension:

T * sin(θ) = m * a

Rotating Body When Friction Acts

Body Rotating on a Horizontal Surface with Friction

ΣF = m * a

F_f + p + N = m * a

Components:

x: F_f = m * a

y: N – p = 0

μ = v² / (R * g)

Rotor

ΣF = m * a

F_f + p + N = m * a

Components:

x: N = m * a

y: p – F_f = 0

F_f = μ * N (N = m * g)

N = m * g / μ

N = m * ω² * R

v = √(g / μ * R)

Peralta Curve

ΣF = m * a

p + N = m * a

Components:

x: N * sin(θ) = m * v² / R

y: N * cos(θ) – m * g = 0

Therefore: tan(θ) = v² / (R * g)