Two equations for centripetal acceleration?

[September24]

A child is on a merry go round and moves towards the center, what happens to centripetal acceleration?

Apparently, the acceleration goes down since radius is decreasing.

A=w^2*r so as radius goes down, so does acceleration. However, how come we don't use the other formula for centripetal acceleration:

A=V^2/R, using this formula, the acceleration goes up.

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[FeinMS]

As the kid walks toward the center, her tangential speed (v) changes. So using a=V^2/R, you can't really see the true effect of R on a because both V and R are changing.

As the kid walks toward the center, her angular velocity (w) doesn't change because at any point on the merry go round, the change in angle displacement over time is the same whether you're at the edge or near the center. Therefore, using a=w^2*r can give you the correct relationship between r and a because w stays the same.

 

[September24]

This may seem like a stupid question, but why is angle dispaclement (used for omega) the same but regular displacement (for velocity) changing.

 

[FeinMS]

Ok so let's imagine that at t=0, the child is at 6 o'clock position of the merry go round, and at t=1, she's at 3 o'clock position. Merry go round is definitely rotating counter clock wise. Also imagine there's a child on the edge and another child near the center.

Between time t=0 and t=1, both children traveled the same angle of 90 degree, however the child on the edge traveled more (linearly) because the arc length of the diplacement is longer for the child on the edge. So linear velocity is larger for the child on the edge. Idk if this explanation is clear or not, so let me know if you need a drawing or something.

 

[September24]

Actually I think I get it. For angular velocity, i think of degrees while for velocity, I think of the displacement along the arc right? I see the the degree is the same for both bodies but one has a larger displacement along their arc. Since angle doesn't change, angular velocity is not so we use the w^2*r equation.

A small follow question, can you think of an occasion where angle DOES change but the displacement along the arc wont. Basically the opposite scenario of the one you described.

 

[FeinMS]

You're right and nope. If the angular velocity is different, that means it's not rotating uniformly as one body. For example, you can have a "double merry go round" where inner disc rotates separately from outer surrounding doughnut shape. Then the two parts will have different angular velocity.

 

[September24]

Thats what I thought. One part of the platform cannot rotate more than the other unless it breaks or something. Thanks for the help.

 

[medemic]

Good thread.