# What are angular momentum and torque?

• erocored

#### erocored

Wikipedia says that they are the equivalents of momentum and force in rotational motion but I don't understand why this comparison is possible. The torque's dimension is N*m it seems like energy. What is this energy? Why angular momentum is not mass times angular velocity?

The torque's dimension is N*m it seems like energy.
Same dimension doesn't imply the same physical quantity. For energy (work) the Force [N] is parallel to the distance [m]. For torque they are orthogonal.

Why angular momentum is not mass times angular velocity?
Because rotational inertia depends on the distance of the mass to the rotation axis, not just on the amount of mass.

Lnewqban, vanhees71 and etotheipi
Well it's sort of by definition, but the motivation is like @A.T. pointed out, that you expect the angular momentum of the same amount of mass spinning at the same angular velocity at a large radius to be greater than at a small radius. For a single particle, define ##\vec{L} = \vec{r} \times \vec{p} = m\vec{r} \times \dot{\vec{r}}##. Then the angular momentum of a rigid body hinged at the origin is just$$\vec{L} = \sum_i m_i \vec{r}_i \times \dot{\vec{r}}_i = \sum_i m_i \vec{r}_i \times (\vec{\omega} \times \vec{r}_i) = \sum_i m_i (r_i^2 \vec{\omega} - (\vec{\omega} \cdot \vec{r}_i) \vec{r}_i)$$with components$$L_a = \sum_i m_i (r_i^2 \omega_a - \omega_b (\vec{r}_i)_b (\vec{r}_i)_a )$$It's now convenient to define the "moment of inertia", with components as follows$$I_{ab} = \sum_i m_i (r_i^2 \delta_{ab} - (\vec{r}_i)_a (\vec{r}_i)_b)$$so that$$I_{ab} \omega_b = \sum_i m_i \left( r_i^2\omega_a - (\vec{r}_i)_a (\vec{r}_i)_b \omega_b \right)$$which is just the same as our ##L_a##, i.e. $$L_a = I_{ab} \omega_b, \quad \text{or} \quad \vec{L} = I\vec{\omega}$$this ##I## is the moment of inertia tensor, which takes the angular velocity vector to the angular momentum vector. Note that ##\vec{L}## is not necessarily even parallel to ##\vec{\omega}##.

erocored and vanhees71
Wikipedia says that they are the equivalents of momentum and force in rotational motion but I don't understand why this comparison is possible.
Momentum is a conserved quantity and force is its rate of transfer. Similarly angular momentum is a conserved quantity and torque is its rate of transfer.

Why angular momentum is not mass times angular velocity?
Because that quantity is not conserved. Nobody cares about it.

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Frigus, erocored, PeroK and 2 others
... Why angular momentum is not mass times angular velocity?
Two rotating objects can have identical mass and still show different resistances to change the magnitudes of their angular velocities.

The object with more mass concentrated near the axis of rotation will have less inertia, and will increase or decrease its angular velocity more quickly under the action of an applied torque, than the object with more mass concentrated far from the axis of rotation.

The concept of moment of inertia considers that characteristic, while the concept of mass does not.
https://en.m.wikipedia.org/wiki/Moment_of_inertia

erocored, PeroK, vanhees71 and 1 other person
It may be useful to consider the the dictionary definition of the word "moment" in this context. $$moment=importance$$ Torque is the ##moment## of force. Angular momentum is the ##moment## of momentum. Moment of Inertia is the ##moment## of the mass.
In each case the ##moment## is calculated using the nearest distance to the chosen axis, because for rotation that is what is important.

Motore
It may be useful to consider the the dictionary definition of the word "moment" in this context. $$moment=importance$$ Torque is the ##moment## of force. Angular momentum is the ##moment## of momentum. Moment of Inertia is the ##moment## of the mass.
In each case the ##moment## is calculated using the nearest distance to the chosen axis, because for rotation that is what is important.
A quick trip to google suggests that the mathematical definition of "moment" is of more importance than the English dictionary definition.

https://en.wikipedia.org/wiki/Moment_(mathematics)

vanhees71 and hutchphd
Perhaps. I found it useful to know the archaic English when teaching it. Otherwise it lacks color.

vanhees71
Wikipedia says that they are the equivalents of momentum and force in rotational motion but I don't understand why this comparison is possible. The torque's dimension is N*m it seems like energy. What is this energy? Why angular momentum is not mass times angular velocity?
Crack a book! Torque is defined in such a way that makes it proportional to angular acceleration for a rigid body. Angular momentum is defined in such a way that makes it conserved in the absence of a net external torque. Any decent college-level introductory physics textbook will cover this in full detail.

erocored and etotheipi
Well, it's not that simple except if you consider the rotation around a body-fixed axis. A typical example is a physical pendulum. In the general case, you need the tensor of inertia. It's the most chalenging subject of the intro mechanics lecture.

etotheipi