Conceptual problem on momentum

[erisedk]

Homework Statement

Does a purely rotating body (something like a disk hinged at a point and rotating with some angular speed) have zero linear momentum?

Homework Equations

The Attempt at a Solution

I believe it will, but I'm not like completely convinced. I tried to draw an analogy with a purely translating body having zero angular momentum, but that's only when the angular momentum is considered about the center of mass, not any random point. But linear momentum isn't defined with respect to points. So, I'm pretty sure that a purely rotating body has zero linear momentum, but I'd just like somebody to confirm it.

 

[Andrew Mason]

Homework Statement

Does a purely rotating body (something like a disk hinged at a point and rotating with some angular speed) have zero linear momentum?

Homework Equations

The Attempt at a Solution

I believe it will, but I'm not like completely convinced. I tried to draw an analogy with a purely translating body having zero angular momentum, but that's only when the angular momentum is considered about the center of mass, not any random point. But linear momentum isn't defined with respect to points. So, I'm pretty sure that a purely rotating body has zero linear momentum, but I'd just like somebody to confirm it.

Linear momentum is defined relative to an inertial frame of reference. The answer to your question (without the parentheses) is: yes, in the inertial frame of reference of the the centre of mass of the rotating body. Essentially, the linear momenta of all the parts of the body at any given moment sum to zero. The answer to your question (with the parentheses) is: not necessarily. If a disk is hinged at a point other than its centre of mass, then the centre of mass will be rotating and the linear momentum of all parts at a given moment will not sum to zero. It will have non-zero linear momentum relative to the inertial frame of reference defined by the centre of rotation, but that linear momentum will keep changing direction.

AM