Q6.3 Injuns! (S)
A wheel of radius 30 cm has eight spokes and rotates at a steady 2.5 revolutions per second about its center. You want to shoot a 20 cm arrow through the wheel without hitting the spokes. |
- Does it matter where (between the axle and the wheel rim) you aim? If so, where is the best location?
- What minimum speed must the arrow have?
Hint
Reveal
Solution
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Solution
- The angle that a spokes sweeps out in a given time is by definition, constant (it’s the angular velocity). Although different points along a spoke move at different speeds (v=rω), the distance a point on a spoke has to cover to rotate one eigth of a revolution increases with increasing distance (s=rθ). The time it takes a spoke to move a distance s is given by [time] = [distance]/[speed], so that t=s/v=rθ/rω=θ/ω, and this is independent of where we aim.
If the arrrow goes as slowly as it can, its head just misses the receding spoke, and its tail is just missed by the advancing spoke. Therefore the time taken for the arrow to pass through must be the time for a spoke to turn θ=1/8th of a revolution. The time is given by
The arrow must go 20 cm in this time, so the minimum speed is v=4 ms-1