AcademyRotational Dynamics
Academy
Torque and Angular Acceleration
Level 1 - Physics topic page in Rotational Dynamics.
Principle
Net external torque sets angular acceleration through moment of inertia.
Notation
\(\sum\tau_z\)
net external torque about the axis
\(\mathrm{N\,m}\)
\(I\)
moment of inertia about the axis
\(\mathrm{kg\,m^{2}}\)
\(\alpha\)
angular acceleration
\(\mathrm{rad\,s^{-2}}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(R\)
radius where a string or rim force acts
\(\mathrm{m}\)
\(T\)
tension
\(\mathrm{N}\)
Method
Derivation 1: Sum torques about the actual axis
The rotational form of Newton's second law uses torques about the same axis used to define \(I\).
Find each torque
\[\tau_z=rF_\perp\]
Choose rotation sign
\[+\alpha\ \text{matches positive}\ \sum\tau_z\]
Rotational law
\[\sum\tau_z=I\alpha\]
Derivation 2: Add constraints only when the geometry requires them
If a string unwinds without slipping, the string acceleration equals the rim's tangential acceleration.
Tangential acceleration
\[a_t=R\alpha\]
No-slip string constraint
\[a=a_t=R\alpha\]
Constant alpha link
\[\omega_f=\omega_i+\alpha t\]
Derivation 3: Connect torque impulse to angular momentum
The torque law can also be integrated over time to find angular impulse.
Torque law
\[\sum\tau=\frac{dL}{dt}\]
Integrate in time
\[\int\sum\tau\,dt=\Delta L\]
Rules
These are the compact results from the method above.
Rotational law
\[\sum\tau_z=I\alpha\]
Tangential link
\[a_t=R\alpha\]
Angular impulse
\[\int\tau\,dt=\Delta L\]
Constant alpha
\[\omega_f=\omega_i+\alpha t\]
Examples
Question
A wheel has
\[I=0.80\,\mathrm{kg\,m^2}\]
A net torque \[4.0\,\mathrm{N\,m}\]
acts. Find \(\alpha\).Answer
Use
\[\alpha=\frac{\sum\tau}{I}=\frac{4.0}{0.80}=5.0\,\mathrm{rad\,s^{-2}}\]
Checks
- Use the moment of inertia about the actual axis.
- Internal torques cancel only when the whole system is chosen.
- String acceleration and rim acceleration match only with no slip.
- A larger \(I\) gives smaller \(\alpha\) for the same net torque.