Newton’s Second Law for Rotation (DP IB Physics: HL): Revision Note

Newton’s Second Law for Rotation

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

$F=ma$

• 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

$\tau =I\alpha$

• This is Newton's Second Law of rotational motion, where:

  • $\tau$ = torque (N m)

  • $I$ = moment of inertia (kg m²)

  • $\alpha$ = 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:  $F=ma$

Torque:  $\tau =Fr$

• Where:

  • $r$ = distance from the axis of rotation (m)

• Combining these equations gives:

$\tau =r\left(ma\right)$

• The moment of inertia of a rotating body can be thought of as analogous to 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:  $\alpha =\frac{a}{r}$

Moment of inertia (point mass):  $I=m{r}^2$

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

$\tau =r\left(mr\alpha \right)$

$\tau =\left(m{r}^2\right)\alpha$

$\tau =I\alpha$

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