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Let's say you're on a roundabout. When you're radially further away from the roundabout would you feel more centrifugal force or less? I feel like intuitively you'd feel less, but from the actual formula it seems to be the opposite case?

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Let's say the roundabout is spinning at a constant angular velocity $\omega$. An object remaining at a particular point on the roundabout a distance $r$ away from the centre of the roundabout will experience a centripetal acceleration of $\omega^2 r$, which means the required centripetal force to enable this acceleration is $m\omega^2 r$.

Since $\omega$ is assumed to be constant here, the centripetal force indeed gets larger with $r$. Hence why you need more force to hold on near the edges of the roundabout than at the centre, giving the roundabout spins at the same angular velocity.

Also, for more intuitive insight, note that the tangential velocity increases as you move further from the centre, even though the angular velocity does not change, $v_{tan.} = \omega r$:

enter image description here

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  • $\begingroup$ So on a sufficiently large disk, those on the outer rim could be moving the speed of light? $\endgroup$
    – Neil
    Apr 21 '16 at 12:52
  • $\begingroup$ According to Newtonian mechanics, yes. However, Newtonian mechanics break down at these relativistic speeds, and we would need to consider relativity instead. Otherwise, you would surpass the speed of light according to this model; not realistic! However, since the question is based off an everyday object like a roundabout, it would be incredibly impractical to consider relativity. Newtonian mechanics and rigid body dynamics is more than enough. $\endgroup$
    – Involute
    Apr 21 '16 at 16:18
  • $\begingroup$ Furthermore, for large enough wheels, you would need to use solid mechanics: the stresses built up by such a large wheel travelling at a large enough speed will cause the wheel to deform, and possibly break. Rigid body dynamics assumes these stresses are insignificant, which is approximately true for smaller wheels travelling at smaller speeds, ignoring these deformations. If you're interested, I could answer this query in response to a question :) $\endgroup$
    – Involute
    Apr 21 '16 at 16:23

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