- #1
johnschmidt
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Homework Statement
Note that the FAQ is not clear if I am required to use LaTeX for formatting formulas. I have not used LaTeX, but I will retype my question if I am violating the rules. Please let me know :-)
Non-extensible cable attached to ceiling. Cable goes down, around a pulley and then back up, then over another pulley and down. At the end of the cable is mass 1. Hanging from the first pulley is mass 2. I am trying to find a formula for the acceleration of M1 as a function of the mass of 1 and 2.
Homework Equations
I have set g = -9.8m/s^2 (positive acceleration is up, negative is down)
I have a free-body diagram for mass 1:
Total force on mass 1: f1 = f1(up) + f1(down)
f1(up) is the force imparted by the cable pulling up
f1(down) = m1 * g
And for mass 2:
Total force on mass 2: f2 = 2 * f2(up) + f2(down)
f2(up) is the force imparted by each of the two cables lifting on the pulley above mass 2.
f2(down) = m2 * g
I set also f2(up) = f1(up) since the same cable that holds mass 1 also goes to the pulley for mass 2. This is my first question - is this assumption of the forces upward correct?
I also say that because of the pulley arrangement the acceleration of mass 1 (a1) = 2 * the acceleration of mass 2 (a2). I say this because the velocity of mass 1 is double the velocity of mass 2 (due to the pulley arrangement).
So we have:
a1 = 2 * a2
f2(up) = f1(up)
a1 = (f1(up) + f1(down)) / m1
a2 = (2 * f2(up) + f2(down)) / m2
Is this correct so far?
The Attempt at a Solution
Then I try to solve for a1 in terms of m1 and m2.
First, solve for f1(up) in terms of a1:
a2 = (2 * f2(up) + f2(down)) / m2
a2 * m2 = 2 * f2(up) + f2(down)
a2 * m2 = 2 * f1(up) + m2 * g
(a2 * m2) - (m2 * g) = 2 * f1(up)
((a2 * m2) - (m2 * g)) / 2 = f1(up)
(((a1 / 2) * m2) - (m2 * g)) / 2 = f1(up)
Now plug this into the original a1 equation:
a1 = (f1(up) + f1(down)) / m1
a1 = ((((a1 / 2) * m2) - (m2 * g)) / 2 + f1(down)) / m1
a1 = ((((a1 / 2) * m2) - (m2 * g)) / 2 + m1 * g) / m1
a1 * m1 = (((a1 / 2) * m2) - (m2 * g)) / 2 + m1 * g
2 * a1 * m1 = (a1 / 2) * m2 - (m2 * g) + 2 * m1 * g
2 * a1 * m1 - (a1 / 2) * m2 = 2 * m1 * g - (m2 * g)
a1 * (2 * m1 - m2 / 2) = g * (2 * m1 - m2)
a1 = (g * (2 * m1 - m2)) / (2 * m1 - m2 / 2)
Now, as a sanity check, I try some examples:
Example 1:
m1 = 1 kg
m2 = 2 kg
(I would expect this to show no acceleration)
a1 = (-9.8 (2 * 1 - 2)) / (2 * 1 - 2 / 2) = 0
good!
Example 2:
m1 = 1.1 kg
m2 = 2 kg
(I would expect this to show a slow negative acceleration as m1 falls to the floor)
(-9.8 (2 * 1.1 - 2)) / (2 * 1.1 - 2 / 2) = -1.96 / 1.2 = -1.63 m/s^2
good!
Example 3:
m1 = 0.9 kg
m2 = 2 kg
(I would expect this to show a slow positive acceleration as m1 rises to the ceiling)
(-9.8 (2 * 0.9 - 2)) / (2 * 0.9 - 2 / 2) = 1.96 / 0.8 = 2.45 m/s^2
good!
Example 4:
m1 = 0.1 kg
m2 = 2 kg
(I would expect this to show a fast positive acceleration as m1 rises quickly to the ceiling)
(-9.8 (2 * 0.1 - 2)) / (2 * 0.1 - 2 / 2) = 17.64 / -0.8 = -22.05
Huh?
I have been through this several (many!) times and cannot figure out what I'm doing wrong. Can anyone offer some insight?
Additionally, with m1 = 0.5 kg the equation produces zero in the denominator, which is probably not right :-/
Thanks very much in advance!
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