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Since I am freshmen I started to learn mechanics first of all, and was a bit confused by this law (despite it is law of nature).

First of all, as I can judge, this law works only if the body has an angular velocity, right? And it is analogue of the conservation law for straightline motion?

But I still cant imagine how it works by examples, can someone help me?

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    $\begingroup$ -1. No research effort. $\endgroup$ – sammy gerbil Feb 23 '17 at 22:38
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First of all, as I can judge, this law works only if the body has an angular velocity, right?

The angular momentum is a vector quantity - it has magnitude and direction - and it depends on some reference point. This means a particle or a collection of particles (a body) may have a non vanishing angular momentum with respect to some point $O$ but at same time have zero angular momentum with respect to another point $Q$.

The particle's angular momentum with respect to $O$ is defined as $$\vec L=\vec r\times \vec p,$$ where $\vec p$ is the linear momentum of the particle with respect to $O$ and $\vec r$ is the vector radius from $O$ to the position of the particle. If you change the point of reference, the vector $\vec L$ changes.

That said, we can see that is not necessary for the particle to rotate along an axis in order to have a non vanishing angular momentum. Even a particle travelling in a straight line can have angular momentum. Its angular momentum will be different from zero when computed with respect to any point which does not lie in the line of its trajectory.

The angular momentum theorem says that the rate of change of the angular momentum of a body equals the net external torque, both computed with respect to the same point. This is a general law and holds for any motion of the body.

And it is analogue of the conservation law for straightline motion?

The angular momentum in rotational dynamics plays the same role as the linear momentum does for straight line dynamics. It contains information about inertia and motion, just like $\vec p$ does. The dynamical law for rotations (as mentioned above) reads, $$\frac{d\vec L}{dt}=\vec\tau.$$ Note it is analogous the the Newton's second law, $$\frac{d\vec p}{dt}=\vec F,$$ where $\vec F$ is the net external force.

But I still cant imagine how it works by examples, can someone help me?

Let me illustrate it with a trivial example. Consider a particle on a frictionless horizontal table. At the initial instant it has velocity along the $x$ direction. The external torque is given by $\vec \tau = \vec r\times \vec F$ and since the net external force vanishes (normal-weight) the torque is zero and the angular momentum must be conserved. Now compute the angular momentum with respect to the origin and show it gives zero for any instant of the motion and therefore is conserved. The next exercise is to compute the angular momentum with respect to a point $Q$ which does not lie in the $x$ axis. Show that $\vec L$ is not zero, but it is still constant.

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The classic example is a figure skater in a spin. Without adding any additional force, by simply pulling in their arms and making themselves more compact, their spin speeds up in a direct application of conservation of angular momentum. Mass remains constant, but the moment of inertia is reduced by the act of pulling in the arms, so the spin velocity must increase to keep angular momentum constant.

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Figure skating spins

If you notice how figure skaters spin, you can see that they rotate faster when they bring their arms closer(decreasing the moment of inertia)

Since the ice surface is almost frictionless and no other torque is applied on the skater, her angular momentum is conserved resulting in faster rotation.

Physics behind figure skating

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When I was at college the library had chairs which could spin freely; sitting on one of them and keeping my arms extended out I used my feet to make the chair spin; then by bringing my arms in it would spin faster!

The physics is that I was reducing my moment of inertia, but given that angular momentum is conserved, the angular velocity must increase; hence the faster spinning chair when I pulled my arms in.

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protected by Qmechanic Feb 23 '17 at 22:42

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