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Kinetics: Force, Mass, and Acceleration

Key Takeaways

  • Newton's Second Law: ΣF = ma — the net force on a body equals its mass times its acceleration.
  • Weight W = mg, where g = 9.81 m/s² (SI) or 32.2 ft/s² (USCS).
  • For angular motion: ΣM = Iα, where I is the mass moment of inertia and α is angular acceleration.
  • The mass moment of inertia depends on the body's mass distribution and the axis of rotation.
  • Parallel axis theorem for mass: I = Ī + md², where d is the distance from the centroidal axis.
  • For a system of particles: ΣF = maG, where aG is the acceleration of the center of mass.
Last updated: March 2026

Kinetics: Force, Mass, and Acceleration

Newton's Laws of Motion

LawStatement
First (Inertia)A body at rest stays at rest; a body in motion continues at constant velocity — unless acted on by a net external force
Second (F = ma)ΣF = ma — net force equals mass times acceleration
Third (Action-Reaction)For every action, there is an equal and opposite reaction

Linear Motion (Particle Kinetics)

∑Fx=max∑Fy=may\sum F_x = ma_x \qquad \sum F_y = ma_y

Steps to solve:

  1. Draw the free-body diagram (all external forces)
  2. Draw the kinetic diagram (ma in the direction of acceleration)
  3. Apply ΣF = ma in each direction
  4. Solve the system of equations

Normal-Tangential Components

∑Ft=mat∑Fn=mv2ρ\sum F_t = ma_t \qquad \sum F_n = m\frac{v^2}{\rho}

Example: A 2,000 kg car travels at 25 m/s around a curve of radius 50 m. What friction force is needed?

ΣFn = mv²/r = 2,000 × (25)²/50 = 2,000 × 12.5 = 25,000 N = 25 kN

Angular Motion (Rigid Body Kinetics)

∑MG=IGα\sum M_G = I_G \alpha

where:

  • ΣMG = net moment about the center of mass
  • IG = mass moment of inertia about the center of mass
  • α = angular acceleration (rad/s²)

For rotation about a fixed axis O: ∑MO=IOα\sum M_O = I_O \alpha

Mass Moments of Inertia (about centroidal axes)

ShapeI
Slender rod (length L)mL²/12
Thin rectangular plate (a × b, about a-axis)mb²/12
Solid cylinder/disk (radius r, about axis)mr²/2
Thin-walled hollow cylinder (radius r)mr²
Solid sphere (radius r)2mr²/5
Thin spherical shell (radius r)2mr²/3

Parallel Axis Theorem (Mass)

I=Iˉ+md2I = \bar{I} + md^2

where d is the distance from the centroidal axis to the parallel axis.

General Plane Motion

For a rigid body in general plane motion (translation + rotation):

∑F⃗=ma⃗G\sum \vec{F} = m\vec{a}_G ∑MG=IGα\sum M_G = I_G \alpha

These equations apply simultaneously — the body translates with its center of mass AND rotates about its center of mass.

Rolling Without Slipping

For a disk or wheel rolling without slipping:

  • Velocity: vG = rω
  • Acceleration: aG = rα
  • Friction: The static friction force at the contact point does NOT do work (no sliding)

Contact point velocity is zero: The point of the wheel touching the ground has zero velocity at that instant.

Test Your Knowledge

A 10 kg block on a frictionless surface is pushed with a force of 50 N. What is its acceleration?

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Test Your Knowledge

What is the mass moment of inertia of a solid disk (mass 5 kg, radius 0.3 m) about its central axis?

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D
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