Pulley attached to a pulley - Find the balance equation

In summary, the conversation discusses finding an equation that relates the masses m1, m2, and m3 in a system involving a fixed pulley and a massless pulley. The goal is to make m3 stationary. After some discussion and equations, it is determined that m1 and m2 must have equal and opposite accelerations in order for m3 to be still. This leads to the equation m3 = 4m1m2 / (m1 + m2).
  • #1
AHashemi
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Homework Statement


A string is hung over a fixed pulley, and a mass of m3 is suspended from one end of the string. The other end of the string supports a massless pulley, over which a second string is hung. This string has a m2 mass attached to one end, and a m1 mass attached to the other end.
Find an equation between m1, m2 and m3 that makes the m3 still.

hw-evil-pulleys.png


Homework Equations


F=ma

The Attempt at a Solution


I took the upward as positive direction of y. and wrote these equations based on F=ma for each mass:
m3: [tex]T_1 - m_3g=m_3 a_3[/tex]
m2: [tex]T_2 - m_2g=m_2 a_2[/tex]
m1: [tex]T_2 - m_1g=m_1 a_1[/tex]

but I can't find out how to make m3 still.
 
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  • #2
If m3 has to be still, what can you say about m3's acceleration? What does this mean with regard to the downward acceleration of the lower pulley? How are the acclerations of m1 and m2 related?

Chet
 
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  • #3
Chestermiller said:
If m3 has to be still, what can you say about m3's acceleration? What does this mean with regard to the downward acceleration of the lower pulley? How are the acclerations of m1 and m2 related?

Chet

emm... well a3 has to be equal to 0. by this we can say [tex]T_1=m_3g[/tex] and we know [tex]T_1=2T_2[/tex] so [tex]2T_2=m_3g[/tex] but I have no idea about how to find left pulley's acceleration which is necessary to find T1.
I've spend too much time on this. I'm sure I'm missing something.
 
  • #4
AHashemi said:
emm... well a3 has to be equal to 0. by this we can say [tex]T_1=m_3g[/tex] and we know [tex]T_1=2T_2[/tex] so [tex]2T_2=m_3g[/tex] but I have no idea about how to find left pulley's acceleration which is necessary to find T1.
I've spend too much time on this. but I'm sure I'm missing something.
If m3 is not accelerating, then the lower pulley, which is joined to m3 by an inextensible string is not accelerating either. So that pulley has to be stationary.

Chet
 
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  • #5
Chestermiller said:
If m3 is not accelerating, then the lower pulley, which is joined to m3 by an inextensible string is not accelerating either. So that pulley has to be stationary.

Chet
Oh right.. that was a mistake.
But I can't understand how m1 and m2 are related to m3's acceleration. their net weight just has to be equal to m3. do their acceleration matter at all?
 
  • #6
AHashemi said:
Oh right.. that was a mistake.
But I can't understand how m1 and m2 are related to m3's acceleration. their net weight just has to be equal to m3. do their acceleration matter at all?
Sure. Their accelerations affect the tension (see your own equations). If the pulley is stationary, how is the upward acceleration of m1 related to the downward acceleration of m2?

Chet
 
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  • #7
Chestermiller said:
Sure. Their accelerations affect the tension (see your own equations). If the pulley is stationary, how is the upward acceleration of m1 related to the downward acceleration of m2?
Chet
[tex]a_1=g(m_2-m_1)/(m_1+m_2)=-a_2[/tex]
they have equal magnitude but opposite directions.
 
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  • #8
So, based on this, what is T2 equal to?
 
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  • #9
Chestermiller said:
So, based on this, what is T2 equal to?
[tex]T_2=m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
so
[tex]T_1=2m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
and because we need T1 to be equal to m3g we can say:
[tex]m_3g=2m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
which is:
[tex]m_3=2m_1((m_2-m_1)/(m_2+m_1)+1)[/tex]
Oh! that's right! also m1=m2=m results m3=2m fits into this!
Thanks!
 
  • #10
AHashemi said:
[tex]T_2=m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
so
[tex]T_1=2m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
and because we need T1 to be equal to m3g we can say:
[tex]m_3g=2m_1g((m_2-m_1)/(m_2+m_1)+1)[/tex]
which is:
[tex]m_3=2m_1((m_2-m_1)/(m_2+m_1)+1)[/tex]
Oh! that's right! also m1=m2=m results m3=2m fits into this!
Thanks!
Good job, but just for aesthetic purposes, why don't you reduce that thing in parenthesis to a common denominator?
 
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  • #11
Chestermiller said:
Good job, but just for aesthetic purposes, why don't you reduce that thing in parenthesis to a common denominator?
[tex]m_3=2m_1(2m_2/(m_2+m_1))[/tex]

Thanks again for your help.
 
  • #12
AHashemi said:
[tex]m_3=2m_1(2m_2/(m_2+m_1))[/tex]

Thanks again for your help.
Isn't that the same as $$m_3=\frac{4m_1m_2}{(m_1+m_2)}$$
 

Related to Pulley attached to a pulley - Find the balance equation

What is a pulley attached to a pulley?

A pulley attached to a pulley is a system of two or more pulleys connected together through a single rope or belt. This allows for a mechanical advantage, making it easier to lift heavy objects by distributing the weight over multiple pulleys.

How do you find the balance equation for a pulley attached to a pulley?

The balance equation for a pulley attached to a pulley involves calculating the tension in the rope or belt and the weight of the object being lifted. The equation is: T1 x D1 = T2 x D2, where T1 and T2 are the tensions on each side of the pulley, and D1 and D2 are the distances the rope or belt travels on each side.

What is the mechanical advantage of a pulley attached to a pulley?

The mechanical advantage of a pulley attached to a pulley is equal to the number of pulleys in the system. For example, if there are two pulleys in the system, the mechanical advantage is 2. This means that the weight being lifted will feel half as heavy as it actually is.

What are the different types of pulleys that can be used in a pulley attached to a pulley system?

There are three main types of pulleys that can be used in a pulley attached to a pulley system: fixed pulleys, movable pulleys, and compound pulleys. Fixed pulleys are stationary and only change the direction of the force. Movable pulleys can move with the weight being lifted, providing a mechanical advantage. Compound pulleys combine fixed and movable pulleys to provide both a change in direction and a mechanical advantage.

What are some real-world applications of a pulley attached to a pulley system?

Pulley attached to a pulley systems are commonly used in lifting and hoisting heavy objects, such as in construction or on ships. They are also used in elevators and cranes to lift and lower loads. In everyday life, pulley attached to a pulley systems can be found in window blinds, flagpoles, and exercise equipment.

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