The Rotational Law of Motion

Section 5.12 The Rotational Law of Motion

Since a rigid object undergoes both translational and rotational motion, it is important to be able to relate the various forces acting on an object to each kind of motion. This is done by creating an angular analogue of The Law of Motion.

Principle 5.12.1. The Rotational Law of Motion (Newton’s 2nd Law for Rotation).

The net torque on an object or system is equivalent to the product of object’s moment of inertia \(I\) and its angular acceleration \(\alpha\text{:}\)

\begin{equation*} \vec{\tau}_{net} = I\vec{\alpha} \end{equation*}

The angular acceleration \(\vec{\alpha}\) introduced in Section 5.4 is the angular analogue for the translational acceleration. The moment of inertia \(I\) is the angular analogue for mass, which you will study in more detail in upcoming sections. The net torque is a direct analogue to the net force: you can find it by determining the vector sum of all torques acting on a system.

Definition 5.12.2. Net torque.

The net torque on an object or system is equal to the (vector) sum of all torques acting on it:

\begin{equation*} \vec{\tau}_{net} = \sum_i \vec{\tau}_i = \vec{\tau}_1 + \vec{\tau}_2 + \vec{\tau}_3 + \dots \end{equation*}

Exercises The Catapult

Recall Figure 5.10.3, for which you previously drew an extended free-body diagram and found the torque exerted by each force.

1.

Use your previous answers along with the Rotational Law of Motion to determine the relationship between \(m_1\) and \(m_2\) that will allow the catapult to launch the rock.

Solution.

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