Question

The wheel shown has eight equally spaced spokes and a radius of 30 cm. It is mounted on a fixed axle and is spinning at 2.5 rev/s. You want to shoot a 20-cm-long arrow parallel to this axle and through the wheel without hitting any of the spokes. Assume that the arrow and the spokes are very thin.

(a) What minimum speed must the arrow have?

(b) Does it matter where between the axle and rim of the wheel you aim? If so, what is the best location?

Answer 1

The arrow travels a distance equal to its own length while passing through the wheel, agree?
Suppose its speed is V then, the time taken to cross is L/V where L is length of arrow
\[t \space = \space \frac{ L }{ V }\]

Answer 2

next, within this time t, the wheel must rotate AT MOST 1/8 th of a full rotation
Else, the spoke will hit the arrow somewhere

Answer 3

so the arrow has t = 20/v seconds to pass through, the best case

Answer 4

Right

Answer 5

Do we know t?

Answer 6

we have to find it i think

Answer 7

Yep. So, read the question again, you will get it ;)

Answer 8

Ahh you're so fast !
it seems the wheel can rotate a maximum of 1/8th of revolution in 20/v seconds, so we have
\[20/v = \dfrac{\pi/4}{2.5*2\pi} \implies v = 400 cm/sec\]

Answer 9

ofcourse you're fast, you have an alien head :D

Answer 10

any ideas on part b

Answer 11

Hahahahahaha

Answer 12

if it were me with the bow and arrow, my intuition tells me to aim closer to the rim so that i have large space to cover... but my intuition is almost always wrong... so il have to think a bit more...

Answer 13

I will aim to the middle to minimize the accidental hitting on sides

Answer 14

If you are a perfect shooter, it does not matter

Answer 15

any location will be equally good

Answer 16

:) yes

Answer 17

Just dont aim at the center where there is the axle

Answer 18

lol we could always aim outside the wheel
but no prize for you, thats not what the question is asking haha

Answer 19

Oh are you saying, it doesn't matter where the arrow is aimed at ?

Answer 20

yes m8

Answer 21

haha only if ur like best at shooting in wrld :D

Answer 22

it does NOT matter

Answer 23

it will matter if m shooting :)

Answer 24

me too, i'll prolly shoot right in your eyeball @imqwerty

Answer 25

oh, oooops

Answer 26

may i know why it wouldn't matter

Answer 27

ahahaha cx

Answer 28

you're one curious man ganeash

Answer 29

i know you're not kidding because the answer on the back of textbook says that it doesn't matter

Answer 30

:O

Answer 31

but let's be real, b/c the location does not enter our equations
The V we calculated does not depend on the radial location

Answer 32

https://s3.amazonaws.com/upload.screenshot.co/dde86c2501

Answer 33

Another way to look at it is this:

As you go farther out along the radius, the rotational velocity increases. However, the available distance along the circumference also increases PROPORTIONATELY.

Answer 34

This basically render the radial position irrelevant

Answer 35

Wow! I completely missed that!
angular velocity is 2.5 rev/s, which is constant, so all the particles on a spoke move by same angle in a given time

Answer 36

yes thas correct explanation :)

what i said was wrng the time taken was equal but not the velocity

Answer 37

;)

Answer 38

This alien is a genius!
making full use of her huge head :P

Answer 39

tru that

Answer 40

I try ma best :P

Answer 41

it's power comes from extracting energy from human brains

Answer 42

LOL

Answer 43

O.o
*stays away from that alien then*

Answer 44

even you can use your brain to its full potential, there is just a small twist which I ain't revealing, uh-nuh

Answer 45

(:

Answer 46

I guess I have to unfan @Miracrown so that I can fan her now.

Answer 47

thinking a bit more, below is so wrong :
```
Wow! I completely missed that! angular velocity is 2.5 rev/s, which is constant,
so all the particles on a spoke move by same angle in a given time
```

i think the actual reason that the placement doesn't matter is the fact that it is a "rigid body". Even if the angular velocity changes, it changes the same for all the particles on the spoke. So constant angular velocity has nothing to do with it.

Answer 48

It is evident that all points on 1 spoke rotate to th epositin of the previous spoke in the same time - other wise the wheel would disintegrate

Answer 49

Exactly!