A particle of mass m moves under the influence of gravity in a circular orbit of radius R and a time period T. At a certain instant, the angular momentum of the particle is brought to zero. Find the time required for it to fall into the centre of the force.
Added by Melissa N.
Step 1
The time period of the orbit is \( T \). At some instant, the particle's angular momentum is suddenly brought to zero, meaning it no longer has any tangential velocity and will start falling radially inward under gravity. Show more…
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The quarter-circular sector of mass $m$ and radius $r$ is set into small rocking oscillation on the horizontal surface. If no slipping occurs, determine the expression for the period $\tau$ of each complete oscillation.
CIM.6 Imagine slicing a thick disk of radius R in half along its diameter: If you stand the half-disk on its curved edge and nudge it, it will rock back and forth. If the rocking is not too extreme, the time T required for a complete back-and-forth oscillation turns out to be nearly independent of the angle through which the disk rocks. The only other things that T might plausibly depend on are the disk's radius R, its mass M, and the local gravitational field strength g (in m/s^2), since gravity is what is causing the rocking motion. (If you think about it, the disk's thickness is only relevant in that a thicker disk has more mass than a thinner one, so we already have this covered if we consider dependence on M.) Use dimensional analysis to find a reasonable formula for this rocking time up to a unitless constant.
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