Rotational Dynamics

This topic is contained in pages $29-51$ of the textbook.

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An unbalanced torque causes a change in the angular (rotational) motion of an object.

The following relationship can be used to carry out calculations involving torque (measured in $Nm$), perpendicular force, distance from the axis.

$$ \tau =Fr $$

→ The direction that the force is applied is also important ⇒ $\tau = Fsin\theta r$

If something is in rotational equilibrium, the torque in one direction equals the torque in the other direction, i.e. $\Sigma \tau = 0$.

If a rotating object is subject to friction, then $\tau_{unbalanced}=\tau_{applied}-\tau_{frictional}$.

When considering the impact of the mass of an object on itself, assume that all the mass is concentrated in the centre of gravity (normally the centre) and calculate the distance of this from the pivot for $r$.

Applied torques will cause angular acceleration about an axis of rotation, transferring rotational kinetic energy to objects. The following relationship can be used to carry out calculations involving work done, torque, and angular displacement.

$$ E_w=\tau \theta $$

→ The equivalent of $E_w=Fd$

The following relationship can be used to carry out calculations involving torque, angular acceleration, and moment of inertia.

$$ \tau=I\alpha $$

The moment of inertia of an object is a measure of its resistance to angular acceleration about a given axis, measured in $kg \space m^2$.

The moment of inertia of an object depends on mass and the distribution of mass about a given axis of rotation.

→ This is an inverse relationship. If the mass of an object is distributed closer to the axis of rotation, then the moment of inertia will reduce, if the mass is distributed farther away from the axis of rotation then the moment of inertia will increase.

The following relationship can be used to determine the moment of inertia for a point mass.

$$ I=mr^2 $$