HL Only: Torque

• Torque $\tau$ of a force about an axis is given by $\tau = Fr\sin\theta$.

• Include the clockwise or counter-clockwise sense when identifying a torque.

• A formal vector treatment of torque is not required.

  

  The diagram visualises the quantities used in $\tau = Fr\sin\theta$. It helps connect the applied force, distance from the axis and angle.

HL Only: Uniform angular acceleration equations

HL Only: Moment of inertia

• Moment of inertia $I$ depends on the distribution of mass about the axis of rotation.

• For a system of point masses, calculate it using $I=\sum mr^2$.

• Each contribution contains $r^2$, so distance from the rotation axis has a squared effect on the contribution to $I$.

• For a specific extended-body mass distribution, the required moment of inertia equation will be provided.

  

  A point mass $m$ lies a distance $r$ from the rotation axis. This visualises the $mr^2$ contribution in $I=\sum mr^2$.

HL Only: Angular impulse

• A resultant torque acting for a time interval produces an angular impulse.

• The change in angular momentum is $\Delta L=\tau\Delta t=\Delta(I\omega)$.

• Relate changes in $I$ and $\omega$ to the resulting $\Delta L$ when a torque acts.

HL Only: Modelling and examination scope

• No centre of mass calculation is required.

• For linear motion, an extended body’s mass may be treated as concentrated at its centre of mass.

• Simultaneous rotational and translational motion is restricted to rolling without slipping.

• A specific moment of inertia equation will be provided when necessary.

• Questions include changes in $I$ for extended bodies and coupled pairs of bodies.

• Use angular speed instead of a formal vector treatment of angular velocity.

HL Only: Rotational equilibrium

• A body is in rotational equilibrium when the resultant torque is zero.

• Clockwise and counter-clockwise torques must combine to give $\tau_{\text{resultant}}=0$.

• An unbalanced torque on an extended rigid body causes angular acceleration.

HL Only: Describing rotational motion

• Rotation is described using angular displacement $\Delta\theta$, angular velocity $\omega$ and angular acceleration $\alpha$.

• For this course, angular speed is used rather than a formal vector treatment of angular velocity.

• The term angular velocity is still used, but formal vector treatment is not required.

  

  The diagram represents rotational motion and angular velocity $\omega$. Use it to visualise angular motion without requiring formal vector analysis.

HL Only: Newton’s second law for rotation

• Newton’s second law for rotation is $\tau=I\alpha$, where $\tau$ is the average torque.

• The equation links the applied torque with the body’s moment of inertia $I$ and angular acceleration $\alpha$.

• Use it for the rotational dynamics of an extended rigid body.

HL Only: Angular momentum and conservation

• A rotating extended body has angular momentum $L=I\omega$.

• Angular momentum remains constant unless the body is acted upon by a resultant torque.

• If the resultant torque is zero, $I\omega$ remains constant.

• Situations may involve changing $I$ in extended bodies and coupled pairs of bodies.

  

  The wheel demonstration illustrates conservation of angular momentum. When no resultant external torque acts, use $L=I\omega$ as the conserved rotational quantity.

HL Only: Rotational kinetic energy

The kinetic energy of rotational motion is $E_k=\dfrac{1}{2}I\omega^2$.

It can also be written as $E_k=\dfrac{L^2}{2I}$.

Choose the form that matches the quantities supplied in the problem.