Pulley problem, my brain hurts!

[Chrisb]

Hi, here’s the problem.
We have a pulley. It’s r=0.74”. The pulley rotates 68degrees.

3.22” away is another pulley.
the second pulley must rotate 90 degrees slaved off the first pulley

what is the r of the second pulley?
any help would be appreciated. I’ve tried looking oNline for help but people seem to focus more on rotational velocity than ratio in terms of distance.

cheers!

 

[tkhunny]

1) Why does it matter how far apart they are?
2) If you're sitting on pulley #1, and you go for a 68º ride, how far did you go?

 

[Deleted member 4993]

Hi, here’s the problem.
We have a pulley. It’s r=0.74”. The pulley rotates 68degrees.

3.22” away is another pulley.
the second pulley must rotate 90 degrees slaved off the first pulley

what is the r of the second pulley?
any help would be appreciated. I’ve tried looking oNline for help but people seem to focus more on rotational velocity than ratio in terms of distance.

cheers!

What must remain constant during the motion of the two given pulleys?

 

[Chrisb]

1) Why does it matter how far apart they are?
2) If you're sitting on pulley #1, and you go for a 68º ride, how far did you go?

I’ve seen a lot of angular equations with pulley distance. Honestly I’m not sure if it matters or not but thought I’d include the measurements.

the circumstance of the first pulley is 4.647”
The distance traveled in 68degrees is 0.883”

not sure what to do next...

 

[Chrisb]

What must remain constant during the motion of the two given pulleys?

Other than they are connected I think that’s it. The second pulley is slaves off the first pulley but traveling further, I know the radius of the second pulley must be smaller, that where I’m stuck.

 

[Deleted member 4993]

Other than they are connected I think that’s it. The second pulley is slaves off the first pulley but traveling further, I know the radius of the second pulley must be smaller, that where I’m stuck.

How do you say that? - angular distance traveled or circumferential distance traveled?

 

[Chrisb]

How do you say that? - angular distance traveled or circumferential distance traveled?

Angular distance seems based on pulley radius. To say it as circumferential distance isn’t really important to me, it’s angular rotation slaved off another pulley as a lever, let’s say, and that the second pulley is a ratio of the first. So as the first pulley rotates 68degrees, the second slaves pulley rotates 90 degrees. The radius of the second pulley should be smaller... But what’s that radius?I can’t find an equation for that

 

[Deleted member 4993]

Angular distance seems based on pulley radius. To say it as circumferential distance isn’t really important to me, it’s angular rotation slaved off another pulley as a lever, let’s say, and that the second pulley is a ratio of the first. So as the first pulley rotates 68degrees, the second slaves pulley rotates 90 degrees. The radius of the second pulley should be smaller... But what’s that radius?I can’t find an equation for that

How are these two pulleys "connected" to each other?

 

[Chrisb]

Both pulleys are fixed 3.22” apart. You could imagine the pulleys as levers but they are pulleys.

 

[Deleted member 4993]

Both pulleys are fixed 3.22” apart. You could imagine the pulleys as levers but they are pulleys.

For one pulley to be "slave" and the other one be driver, those must be connected some way - may be with an "in-extensible" chain or string. You CANNOT consider pulleys similar to levers - those are different animals.

Now tell us - how are your pulleys connected?

 

[Chrisb]

Ok yes, sorry let me clarify. The pulleys are connected by an in-extensable string.

 

[Deleted member 4993]

Ok yes, sorry let me clarify. The pulleys are connected by an in-extensable string.

Thus the circumferential distance traveled in each pulley would be equal.

\(\displaystyle S = r_1 * \theta_1 = r_2 * \theta_2\)

That 3.22" is a red-herring!

 

[Chrisb]

So, in my case, pardon me- I don’t have symbols on my phone...
S = same in either case
r1 is .74
Theta is 68 degrees (1.18 rad)
=
r2 is ?
Theta is 90 degrees (1.57
So r2 = .98 ?

But that tells me the master pulley of a lessor diameter is driving a bigger pulley a bigger angle. How can this be? I’m doing something wrong.

 

[Steven G]

What is r₁*θ₁? Isn't it .74*1.18 = .8732

So r₂*pi/2 = .8732. Solve for r₂.

 

[Deleted member 4993]

So, in my case, pardon me- I don’t have symbols on my phone...
S = same in either case
r1 is .74
Theta is 68 degrees (1.18 rad)
=
r2 is ?
Theta is 90 degrees (1.57
So r2 = .98 ?

But that tells me the master pulley of a lessor diameter is driving a bigger pulley a bigger angle. How can this be? I’m doing something wrong.

Converting all the degree measures of the angles to radians is the correct thing to do - but in this problem you don't "have to" do that.

r₁*θ₁ = r₂*θ₂

0.74 * 68 = r₂ * 90

r₂ = 0.74 * 68/90 = 0.5591 = 0.56"

 

[Chrisb]

Okayyy I think i understand. I will apply this to my problems. Thank you sooooo much! I really appreciate the help!