# Atwood's Machine with unknown masses

## Homework Statement

A simple Atwood's machine uses two masses m1 and m2. Starting from rest, the speed of the two masses is 8.7 m/s at the end of 7.4 s. At that time, the kinetic energy of the system is 61 J and each mass has moved a distance of 32.19 m.

Find the value of the heavier mass.

Find the value of the lighter mass.

## Homework Equations

a = (m2 - m1) / (m1 + m2)g
Ke = 1/2mv2

## The Attempt at a Solution

61 = 1/2m1v2 + 1/2m2v2
61 = 1/2(8.7)2(m1+m2)
61 = 37.845(m1+m2)
m1+m2 = 1.612

a = 8.7 / 7.4 = 1.176

1.176 = (m2 - m1) / (m1 + m2)g
10.976m1 = 8.624m2
m1 = .786m2

.786m2 + m2 = 1.612
2m2 = 2.051
m2 = 1.025

m1 + 1.025 = 1.612
m1 = .587

I was sure that I solved it correctly, perhaps my algebra for solving the acceleration equation is incorrect.

Thanks for any help.

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.786m2 + m2 = 1.612
2m2 = 2.051
-------> Shouldn't this be 1.786m2=1.612?

-------> Shouldn't this be 1.786m2=1.612?
So you are saying my algebra for solving the acceleration equation is wrong? I got .786, not 1.786.

So you are saying my algebra for solving the acceleration equation is wrong? I got .786, not 1.786.
It seems to me so.
0.786m2 + m2 = 1.612
(0.786+1)m2=1.612
1.786m2=1.612
Could you write down exact solution?

I got .786 from this:

1.176 = (m2 - m1) / (m1 + m2)g
10.976m1 = 8.624m2
m1 = .786m2

the problem is: how do you get from

$$.786m_2 + m_2= 1.612$$

to

$$2m_2 = 2.051$$

$$1.176 = \frac {m_2 - m_1} {m_1 + m_2} g$$

$$10.976m_1 = 8.624m_2$$

this step is also wrong

Last edited:
the problem is: how do you get from

$$.786m_2 + m_2= 1.612$$

to

$$2m_2 = 2.051$$
I divided 1.612 by .786 then I thought I could combine like terms.

$$1.176 = \frac {m_2 - m_1} {m_1 + m_2} g$$

$$10.976m_1 = 8.624m_2$$

this step is also wrong
I was kinda finicky on this step. Could you give me a hint on how to solve it algebraically?

I divided 1.612 by .786 then I thought I could combine like terms.
you have to divide ALL terms by .786. THat would give you $m_2 + m_2/0.786 = 1.612/0.786$ I don't think that helps.

I was kinda finicky on this step. Could you give me a hint on how to solve it algebraically?
[/QUOTE]

multiply both sides by $\frac {m_1 + m_2} {g}$ (you already know what m_1+m_2 is)

combine what you get with $m_1 + m_2 = 1.612$

you have to divide ALL terms by .786. THat would give you $m_2 + m_2/0.786 = 1.612/0.786$ I don't think that helps.
multiply both sides by $\frac {m_1 + m_2} {g}$ (you already know what m_1+m_2 is)

combine what you get with $m_1 + m_2 = 1.612$[/QUOTE]

Okay, so I did this:

1.176 = (m2 - m1) / (m1 + m2) g
1.176 (m1 + m2) g = m2 - m1
1.176 (15.7976) = m2 - m1
18.578 = m2 - m1

I don't know if this is correct because I multiplied m1 + m2 and g. I wasn't sure if I should divide by g or multiply since it is in the denominator.

multiply both sides by $\frac {m_1 + m_2} {g}$ (you already know what m_1+m_2 is)

combine what you get with $m_1 + m_2 = 1.612$
Okay, so I did this:

1.176 = (m2 - m1) / (m1 + m2) g
[/QUOTE]

note that it's $$a = \frac {m_2 - m_1} {m_1 + m_2} g$$

not

$$a = \frac {m_2 - m_1} {(m_1 + m_2) g}$$

so you have to multiply both sides by m_1 + m_2 and then divide by g

Okay, so I did this:

1.176 = (m2 - m1) / (m1 + m2) g
note that it's $$a = \frac {m_2 - m_1} {m_1 + m_2} g$$

not

$$a = \frac {m_2 - m_1} {(m_1 + m_2) g}$$

so you have to multiply both sides by m_1 + m_2 and then divide by g
Okay, here's what I came up with.

1.176 = ((m2 - m1) / (m1 + m2))g

(1.176 * 1.162) / g = .19344 = m2 - m1

How does that look? I do not know what you mean when you say add it to m1 + m2 = 1.612.

Thanks for the help.

if you have A = B and C=D then A+C = B+D

now you have .19344 = m2 - m1

and m1 + m2 = 1.162

if you have A = B and C=D then A+C = B+D

now you have .19344 = m2 - m1

and m1 + m2 = 1.162
A = .19344
B = m2 - m1
C = 1.162
D = m1 + m2

.19344 + 1.162 = (m2 - m1) + (m1 + m2)

How does that set-up look?

Looks OK. you can now add up the right and the left side of this equation.

Looks OK. you can now add up the right and the left side of this equation.
.19344 + 1.162 = (m2 - m1) + (m1 + m2)

1.35544 = m2 + m1

m1 = m2 - 1.35544

Using the equation from Ke and Pe:

m1 + m2 = 1.612

m2 - 1.35544 + m2 = 1.612

m2 + m2 = 2.96744

2m2 = 2.96744

m2 = 1.48372

So m1:

m1 + 1.48372 = 1.612

m1 = .12828

Is my math okay?

.19344 + 1.162 = (m2 - m1) + (m1 + m2)

1.35544 = m2 + m1

m1 = m2 - 1.35544
the only reason for adding the 2 equations was that m1 would disappear.

m1 - m2 + m1 + m2 = ?

if you have 2 equations like 2x - y = 7 and 3x + y = 4 it's often easy to eliminate one
of the variables by adding or subtracting them.
if you add them you get 2x - y + 3x + y = 7 + 4 so you get 5x = 11 and x = 11/5

then you substitute x = 11/5 in 2x - y = 7 to get y: 2 (11/5) - y = 7 => y = 22/5 -7 = 22/5 - 35/5 = - 13/5 = -2.6

adding the equations works because y appears in one of them and -y in the other, so you
know y will disappear.

I understand now, thanks. I also went to my professor to discuss this question and he explained an easier way to do it, but similar.

Thanks for all the help.