Why velocity can change when angular momentum is conserved?

[CollinsArg]

Why the tangential velocity of a particle increase if there are no external torque acting on it and its angular momentum is conserved?

I know that L = I.ω (angular momentum equals moment of inertia times angular velocity)

and v = ω.r (tangential velocity equals angular velocity times the position of the particle), then ω = v/r
doing substitution ⇒ L = I.v/r

Also I know I = m.r² (supposing for one particle, the mass of the particle times its position)

Then L = m.r².v/r ⇒ L = m.r.v

Because the angular momentum is conserved v = L/m.r

Hence, as I change the position of the particle (the same as the radius of the circumference) its velocity changes without any torque being applied, why is it so? Shouldn't velocity be always constant? What does make its velocity change if I can see only a centripetal force (and I learned that centripetal forces can only change the direction of the vector velocity)? Thanks.

 

[Drakkith]

Hence, as I change the position of the particle (the same as the radius of the circumference) its velocity changes without any torque being applied, why is it so? Shouldn't velocity be always constant? What does make its velocity change if I can see only a centripetal force (and I learned that centripetal forces can only change the direction of the vector velocity)?

I believe that the force that is causing the change in position is responsible. For example, an ice skater pulling her arms in while in a spin exerts a force on her arms to bring them in.

 

[CollinsArg]

I believe that the force that is causing the change in position is responsible. For example, an ice skater pulling her arms in while in a spin exerts a force on her arms to bring them in.

Wouldn't it be a centripetal force?

 

[Dale]

I learned that centripetal forces can only change the direction of the vector velocity

Any arbitrary force can be broken into a component parallel to the velocity and a component perpendicular to the velocity. The perpendicular component will change the direction and the parallel component will change the speed.

In uniform circular motion the centripetal force is always perpendicular to the velocity, so it does not change the speed. But a central force in other types of motion (e.g. Orbital motion) will not always be perpendicular. In such cases a central force will change the speed too.

 

[CollinsArg]

Any arbitrary force can be broken into a component parallel to the velocity and a component perpendicular to the velocity. The perpendicular component will change the direction and the parallel component will change the speed.

In uniform circular motion the centripetal force is always perpendicular to the velocity, so it does not change the speed. But a central force in other types of motion (e.g. Orbital motion) will not always be perpendicular. In such cases a central force will change the speed too.

Wound't it mean also that she could slow down the tangential velocity too? depending on the way the ice skater pulled her arms in? But the equation tells me that always when the distance is shortened the tangential velocity increases.

 

[Drakkith]

Wouldn't it be a centripetal force?

Hmm. I'm not sure to be honest. This isn't an area I'm very familiar with.

 

[Dale]

Wound't it mean also that she could slow down the tangential velocity too? depending on the way the ice skater pulled her arms in? But the equation tells me that always when the distance is shortened the tangential velocity increases.

First, simplify the scenario, e.g. A point mass on the end of a string whose length changes. Second, calculate the parallel and perpendicular components of the force, and apply what you know. See if you can identify in what circumstances the speed increases and why.

 

[rcgldr]

Continuing with the point mass on a string on a frictionless surface and a hole that the string can be pulled into or released from. Note that as the string is pulled into or released out of the hole, the path of the mass is spiral like and no longer perpendicular to the string. There's a component of tension that is in the direction of the path of the mass, so the speed of the mass changes.

As an example where the speed does not change, imagine that the string is wrapped around a post of some non-zero radius. The path of the mass will be involute of circle and the string will always be perpendicular to the path of the mass. There is a torque on the post exerted by the string, so angular momentum is not conserved unless you include the post and whatever the post is attached to (like the earth).

Image of the hole case. The short lines are perpendicular to the path, and not lined up with the string:

Image of the post case. The string is perpendicular to the path.