Finding tension and acceleration in a 3 weight, 3 pulley system

[SovXietday]

Homework Statement

This is a "DOE" (Design of Experiment) problem. Basically, the construction is fairly simple. 3 pulleys, 2 on either side, and a free pulley with a weight suspended on it in the middle. Picture is available (don't mind my crappy paint skills, rope lengths are all parallel).

I basically need to compare theoretical velocity of C after falling 5cm and after falling 85cm to my actual measured velocities. However, to be honest my experimental values are all sorts of messed up, and I'm simply not sure if my math is anywhere near right.

m1 = .270kg
m2 = .522kg
m3 = .300kg

Homework Equations

F(net) = ma
xf = 1/2at^2 + Vot + xi (dx/dt and dv/dt respectfully)

The Attempt at a Solution

First I set up my three equations.

m1g - T1 = m1a
T1 = m1g - m1a Eq1

2T2 - m2g = m2a
T2= 1/2(m2g + m2a) Eq2

m3g - T3 = m3a
T3 = m3g - m3a Eq3

So, Fnet = ma should be
T1 + T3 - T2 = ma (what is m? The mass sum of the system?)

Plug in.
m1g - m1a + m3g - m3a - 1/2m2g - 1/2m2a = ma
m1g + m3g - 1/2m2g = a(m + m1 + m3 + 1/2m2)

Isolate for a = (g[m1 + m3 - 1/2m2]) / (m + m1 + m3 + 1/2m2)

If I plug in what I *think* m is supposed to equal (1.092), I get a = 1.923m/s

Integrate a
Vf = 1.923t + Vo (Vo is 0)
Integrate vf
xf = .9615t^2 + xo (Xo is also 0)

Plug in .85m for Xf (furthest displacement of weight 3)
t = .94s

Vf = 1.923(.94)
Vf = 1.808 m/s at .85m

...not even close am I?

 

[ehild]

Do you know the mass of the pulleys? Or they should be assumed massless when calculating the theoretical velocities?

ehild

 

[SovXietday]

Masses of the pulleys are all 7grams. The mass of weight 2 already has the the entire pulley assembly + hanging weight added up. This is a real test, so everything applies.

Real problem is that the teacher has assigned this test for dynamics chapter 1 - rectilinear motion. That's fine if we have distances and velocities, but we don't "know" forces yet, so I'm trying to source all of these equations from backlogs of long ago physics homework and online teachings. :/

 

[ehild]

Which mass is C that falls 5 cm and 85 cm?

The problem is relatively easy if you can ignore the rotational inertia of the pulleys. It is rather complicated otherwise.

In the former case, the tension is the same in all pieces of the rope.
The acceleration of the masses are not the same, however. You need to find out how the acceleration of mass 1 and mass 3 are related to the acceleration of mass 2. Mass 2 is attached to two pieces of the rope. If mass 1 moves downward by Δy1 and mass 3 moves downward by Δy3, mass 2 must move upward by Δy2=(Δy1+Δy3). Take these into account in your equations. You get 3 equations with the unknown T and a1, a3.

ehild

 

[asteriatic]

I would first check to see if there are any errors in the experimental setup or data collection process that could be causing the discrepancies between the theoretical and measured values. It's important to make sure all measurements are accurate and precise.

If the setup and data collection seem to be correct, I would then revisit my calculations to see if there are any errors or simplifications that could be causing the discrepancies. It's possible that there are additional forces or factors that were not accounted for in the equations.

I would also consider using a computer simulation or modeling software to simulate the system and compare the results to the experimental data. This could help identify any potential sources of error and provide a more accurate understanding of the system.

Additionally, I would consult with other scientists or experts in the field to get their input and perspective on the problem. Collaborating and discussing with others can often lead to new insights and solutions.

In conclusion, it's important to carefully analyze all aspects of the problem and seek outside input in order to accurately determine the tension and acceleration in a 3 weight, 3 pulley system.