11.1.5 Newton’s Second Law for Rotation

• In linear motion, the force required to give an object a certain acceleration depends on its mass

• This is Newton's Second Law of linear motion, where:
  • F = force (N)
  • m = mass (kg)
  • a = linear acceleration (m s⁻²)

• In rotational motion, the torque required to give a rotating object a certain angular acceleration depends on its moment of inertia

• This is Newton's Second Law of rotational motion, where:
  • = torque (N m)
  • = moment of inertia (kg m²)
  • = angular acceleration (rad s⁻²)

Newton's second law for rotating bodies is equivalent to Newton's second law for linear motion

• This equation comes from the fact that torque is the rotational equivalent of force:

Force:  

Torque:  

• Where:
  • = perpendicular distance from the axis of rotation (m)

• Combining these equations gives:

• The moment of inertia of a rotating body can be thought of as analogous to (the same as) mass
  • The inertia of a mass describes its ability to resist changes to linear motion, which is referring to linear acceleration
  • Similarly, the moment of inertia of a mass describes its ability to resist changes to rotational motion, which is referring to angular acceleration

Angular acceleration:  

Moment of inertia (point mass): 

• Using these equations with the equations for force and torque leads to:

Comparison of linear and rotational variables in Newton's Second Law

Linear variable	Rotational variable
Force,	Torque,
Mass,	Moment of inertia,
Acceleration,	Angular acceleration,
Newton's Second Law,	Newton's Second Law,