What Is the Minimal Velocity for an Arrow to Pass Through a Rotating Wheel?

[Dansuer]

Homework Statement

A wheel has six spokes of length 30 cm. The wheel is rotating with an angular velocity of 2.5 rotation per second. We want to throw an arrow, of length 24 cm, through the wheel. What is the minimal velocity required for the arrow to go through the wheel without touching the spokes.

Homework Equations

v = s / t

[tex]\omega[/tex] = [tex]\theta [/tex] / t

The Attempt at a Solution

The times for the arrow to go through have to be equal the time one spoke goes around 1/6 of the wheel.

so the system: v = s/t and [tex]\omega[/tex] = [tex]\theta [/tex] / t

gives: v = ([tex]\omega[/tex] / [tex]\theta [/tex]) * s

[tex]\omega[/tex] = 2.5 rotation per second * 360degree = 900 degree/s

[tex]\theta [/tex] = 360 degree / 6 = 60 degree

v = ([tex]\omega[/tex] / [tex]\theta [/tex]) * s = (900 degree/s / 60 degree) * 24 cm = 360 cm

is this right?

 

[cupid.callin]

first, speed is not in cm.

second, convert 24 cm into m

rest is perfectly correct

 

[tiny-tim]

Hi Dansuer!

(have a theta: θ and an omega: ω )

Yes, that looks fine, except for the units …

your speed should be 360 cm/s, not 360 cm, shouldn't it?

(since the question is entirely in cm, I suspect it may not be necessary to convert to m/s)

However, since the arrow is shorter than the radius, you can just throw the arrow gently sideways! ​

 

[Dansuer]

thanks a lot for your answers !

 

[rwisz]

Yes, your solution is correct. The minimal velocity required for the arrow to go through the wheel without touching the spokes is 360 cm/s. This can also be confirmed by calculating the angular velocity of the arrow, which would be the same as that of the wheel (2.5 rotation per second), and then using the equation v = r\omega to calculate the linear velocity of the arrow at the tip, which would also be 360 cm/s.