How do lever arms affect the velocity of objects?

[NA19]

I need help with this attached question! The answer is supposedly B because of something to do with lever arms. But I don't get it at all!

 

[berkeman]

I need help with this attached question! The answer is supposedly B because of something to do with lever arms. But I don't get it at all!

Welcome to the PF.

In problems like this, it is generally best to start by drawing a free body diagram (FBD) showing the tensions...

 

[NA19]

I tried that, but I am not sure how the pulleys and the arrow would affect the forces acting on the strings. My initial instinct was that the tension in A is greater than the tension in B in position 1 because A seems more stretched.

Then, I thought that the tension is the same in position 2 which seems like a rest position.

However, there is no answer like that.

 

[CWatters]

Take a look at the top pulley. Ignore everything else. The pivot is not in the middle.

So..

Q1) The pulley isn't moving so what does that say about the total torque on the pulley?

Q2) Given the answer to Q1... If one force (B) is applied close to the pivot and the other (A) is applied further away from the pivot what does that mean for the forces?

 

[NA19]

Why isn't the pivot in the middle?
Why isn't the pulley moving?

 

[CWatters]

Why isn't the pulley moving?

The first drawing shows the bow in the drawn position ready for firing. In that position the pulley is stationary.

Why isn't the pivot in the middle?

Hint: What do levers do? If B moves a short distance does A move the same distance? More? Less? Same speed as B? Faster? Slower?

 

[NA19]

Q1) The total toque on the pulley is zero.
Q2) Forces farther away are less than when they are close.

I kind of understand what is happening right now, but a few more questions:
How do you determine that this is a lever question instead of a pulley question? I don't really understand how this system acts as a system of levers.
Isn't B farther away from the pivot point than A?

 

[haruspex]

How do you determine that this is a lever question instead of a pulley question? I don't really understand how this system acts as a system of levers.

A pulley can be thought of as a lever, at any given instant. A centrally mounted pulley isn't a very interesting lever because it's 1:1; the distance from fulcrum to contact point is the same for both ends of the contact. If the pulley is eccentrically mounted (and sufficiently frictional that the string does not slide) then it will behave like a lever, except that as the pulley turns the lever ratio will change.
To see this, cut out all parts of the pulley except the two radii to the two ends of the contact.

 

[NA19]

Okay, but...

Isn't B farther away from the pivot point than A in position 1?
Why are we ignoring everything but the top pulley.

 

[haruspex]

Isn't B farther away from the pivot point than A in position 1?

No. The pivot point is the dot, yes? In position 1, string B extent is closer to the dot where it reaches the pulley than is string extent A.

Why are we ignoring everything but the top pulley.

Bottom pulley is the same, by symmetry. Only the pulleys separating extent A from extent B are relevant.

 

[CWatters]

Q2) Forces farther away are less than when they are close.

Levers also change velocities. If B moves slowly what happes to A ?

In case it helps.. The string doesn't slip on the pulley so (for small angles of rotation) you can replace the pulley with lever arms like this. In practice the pulley rotates through quite large angles so levers aren't used but the principle is the same..