What Is the Minimum Speed an Arrow Must Have to Pass Through a Spinning Wheel?

[chaotixmonjuish]

A wheel has eight spokes and a radius of 37.5 cm. It is mounted on a fixed axle and is spinning at 3.75 rev/s. You want to shoot a 17.0 cm long arrow through the wheel, parallel to this axle, without hitting any of the spokes. Assume that the arrow and the spokes are very thin and evenly spaced. What minimum speed must the arrow have?


I just riffled off a few random calculations right away:

radian between the spokes = .7853 radians
radial velocity = .033 rad/sec

I'm not sure what to do after that

 

[Astronuc]

One needs to find the angular velocity (rad/s), which is 3.75 rev/s * 2pi rad/rev. From the angular velocity and the angle between the spokes, one finds the time that the area is open for the arrow to traverse the plane of the wheel.

In order for the arrow to traverse, the full length of the arrow must pass the plane of the spokes in slightly less time than it takes the next spoke to reach the point where the initial spoke passes just before the tip of the arrow reaches the plane of the spokes.

 

[chaotixmonjuish]

So the angular velocity is 3.75*2pi or 23.5619 rad/s. I'm not sure what to do after that. Would i take the angle between and divide it by the angular velocity?

 

[Astronuc]

So the angular velocity is 3.75*2pi or 23.5619 rad/s. I'm not sure what to do after that. Would i take the angle between and divide it by the angular velocity?

Yes - that will give the time between spokes, and that is the time that the length of the arrow must traverse the plane of the wheel (spokes).

 

[chaotixmonjuish]

.7853/23.5619 = .007215 s

would i then use change in radian/change in time

.7853/.007215 to get omega (108.52)

then would i multiply that by the radius (108.52*.375) to get 40.819 m/s