Newton’s Second Law for Rotation (AQA A Level Physics)

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