- #1
Jimmy87
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Homework Statement
A child stands near the middle of a roundabout that is rotating with some angular velocity w. The child moves towards the edge of the roundabout in a straight line from the child's perspective and at constant speed. Explain in as much detail as you can (and using equations) the motion of the child from both the child's perspective and the inertial reference frame. Use Newton's Laws in both cases and comment on the need to introduce fictitious forces in the rotating reference frame.
Homework Equations
a = v^2/r F = mv^2/r
The Attempt at a Solution
Inertial reference frame - sees child's motion as a spiral. Forces would be centripetal force and a tangential force both of which are friction. As the child steps further out their speed must increase because all points along the line they walk must complete a cycle in the same time so the further out you go the bigger the circumference and the higher the speed needs to be. This is provided by a frictional force as they step further out which acts along the tangent. The centripetal force is also fraction towards the centre.
Rotating frame - child sees they are walking in a straight line. Must still exert a frictional force tangent to the path as they move out and towards the centre the whole time. Newton's laws fail here because the child is walking in a straight line at constant speed yet has to exert two resultant forces in order to do this. Coriolis and centrifugal forces must be introduced otherwise Newton's laws would fail.
Bits I am stuck with:
How does the centripetal acceleration change as you go near the edge? At first I though that since the velocity is increasing then so must the centripetal acceleration but the radius is not constant so I am unsure how exactly the centripetal force changes. The radius increases which would decrease the centripetal acceleration but the velocity is increasing.
Something also seems paradoxical to me so I must be missing something. If you are near the middle of a roundabout then to walk to the edge you must have a resultant force towards the edge otherwise you couldn't move. However, the roundabout is constantly in motion so the resultant force must always be to the centre? What am I missing?
Thanks in advance