Acceleration of block 1 in a pulley system

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In summary, the conversation discusses how to find the acceleration of block 1 given the masses of block 1 and block 2, and ignoring friction and pulley mass. The solution involves drawing free body diagrams and using equations to find the acceleration of both blocks. The answer is ultimately found to be 4.4 m/s². The conversation also discusses the relationship between the distances and movements of the two blocks due to the pulley system.
  • #1
ShizukaSm
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Homework Statement


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Find acceleration of block 1, knowing that the mass of block 1 is 2kg and of block 2 is 4kg. You may ignore friction and pulley mass.

Homework Equations



Fr = ma

The Attempt at a Solution



I Drew free body diagrams to block 1, 2 and the pulley, then made the equations:
[itex]
\\
\sum F_{r2} = m_2a_2\\
P_2 - T_2 = m_2a_2\\
\\
\sum F_{rpulley} = m_{pulley}*a\\
2T_2 = T_1\\
\\
\sum F_{1}=m_1a_1\\
T_1=m_1a_1\rightarrow T_2 = \frac{m_1a_1}{2}

\\\\a_1=2a_2 \rightarrow T_2=\frac{m_1(2a_2)}{2} = m_1a_2

\\(Substituting\ in \ first\ equation)\\
P_2-(m_1a_2)= m_2a_2\\
a_2(m_2+m_1) = m_2*g\\\\
a_2=\frac{m_2g}{m_2+m_1} = 6.53\frac{m}{s^2}\rightarrow a_1 = 13.06\frac{m}{s^2}
[/itex]
The answer, however, is supposed to be 4.4 m/s²


Thanks in advance!
 
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  • #2
Is ##a_1 = 2 a_2## true?
 
  • #3
Well, I think it is, isn't it?

If 1 moves a distance "d", 2 will move only d/2 because the rope has to run on both sides of the pulley.
 
  • #4
ShizukaSm said:
Well, I think it is, isn't it?

If 1 moves a distance "d", 2 will move only d/2 because the rope has to run on both sides of the pulley.

On the other hand, if 1 moves a distance "d", the pulley "takes" a length "d" of rope from both sides of the pulley. This rope can only come from the free end holding m2.

attachment.php?attachmentid=54631&stc=1&d=1357924275.gif
 

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  • #5
Wow, that... that makes a lot of sense, so in fact a1 = 2a2

Thanks a lot, gneill!
 

What is a moving pulley?

A moving pulley is a simple machine that is used to lift heavy objects. It consists of a wheel with a groove around its circumference, over which a rope or belt is passed. One end of the rope is attached to the object to be lifted, while the other end is pulled downwards. The pulley rotates, allowing the object to be lifted with less force.

How does a moving pulley work?

A moving pulley works by distributing the force needed to lift an object over multiple ropes or belts. The more pulleys there are, the less force is needed to lift the object. This is because the weight of the object is divided equally between the ropes, reducing the overall force needed to lift it.

What are the advantages of using a moving pulley?

There are several advantages to using a moving pulley. Firstly, it reduces the amount of force needed to lift heavy objects, making it easier for humans to lift them. Additionally, it allows for a more efficient use of energy, as the weight of the object is shared between multiple ropes. Lastly, moving pulleys can be used to change the direction of the force, making it easier to lift objects in difficult or confined spaces.

Are there any limitations to using a moving pulley?

While moving pulleys can make lifting heavy objects easier, they do have some limitations. The weight of the object being lifted must be less than the combined weight of the ropes and pulleys. Additionally, the ropes and pulleys must be strong enough to handle the weight of the object, otherwise they may break under the strain.

How is a moving pulley different from a fixed pulley?

A moving pulley differs from a fixed pulley in that it is able to move along with the object being lifted. In a fixed pulley, the rope is attached to a stationary point, while in a moving pulley, the rope is attached to the object being lifted. This allows for a greater reduction in force needed to lift the object, as the weight is distributed over multiple ropes instead of just one.

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