S5.7 Angular momentum
[A] Definition
- Like all other linear quantities, linear momentum has its angular counterpart: angular momentum.
- Angular momentum is a vector with units kg m2 s-1.
Key Point 5.16
The angular momentum of a particle with respect to a point O is![]()
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[B] Angular momentum of a rigid body
Key Point 5.17
The magnitude of the angular momentum of a rigid body rotating about an axis with angular velocity ω is
![\[ | \vec{L} | = I \omega \]](mastermathpng-2.png)

Note: the moment of inertia depends on the rotation axis.
[C] Newton’s 2nd law for rotation
- It is possible to formulate Newton’s 2nd law in rotational form.
- By considering the total angular momentum of a set of particles about a given point, and using Newton’s third law (Key Point 2.3) to eliminate internal forces, one obtains the general expression relating torque and angular momentum.
Key Point 5.18
Newton’s 2nd Law: angular form relates
the total external torque, , acting on a system of
particles, to the total angular momentum
of the
system about the same point.
![\[ \vec{\tau}_{ext} = \frac{d \vec{L}}{d t} \]](mastermathpng-6.png)
and
are expressed with respect to the same origin.
[D] Special case: Newton’s 2nd law for rigid bodies
For the special case of a rigid body rotating about an axis, Newton’s second law takes a particular form involving the moment of inertia.Key Point 5.19
Newton’s 2nd law for rigid bodies: A torque, τ, applied to a rigid body rotating about a fixed axis produces an angular acceleration α about that axis
![\[ \tau = I \alpha = I \frac{d\omega}{dt} = I \frac{d^2\theta}{dt^2} \]](mastermathpng-9.png)
where I is the moment of inertia about the rotation axis, ω the angular velocity and θ the angular displacement.
Learning Resources
![]() | HRW Chapter 10.9. 11.7-11.10 |
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