Acceleration of Two Masses on a Pulley with Moment of Inertia

  • Thread starter Karol
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In summary, the pulley in the drawing has a moment of inertia of I=kMR2 and the question is asking for the acceleration. Using the equations M=I\alpha and F=ma, the attempt at a solution involved setting up equations for the forces acting on the pulley and using the relationship R\alpha=a to find the acceleration. However, there was a mistake in the second equation, where the g term was missing. After correcting this error and solving for a, the correct answer is a=\frac{g(m_1-m_2)}{kM+m_1+m_2}.
  • #1
Karol
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Homework Statement


The pulley on the drawing has moment of inertia of I=kMR2.
What's the acceleration.

Homework Equations


[tex]M=I\alpha[/tex]

[tex]F=ma[/tex]

The Attempt at a Solution


[tex]m_1g-T_1=m_1a[/tex]
[tex]T_2-m_2=m_2a[/tex]
[tex]\mbox{ }R\alpha=a[/tex]
[tex](T_1-T_2)R=kMR^2\cdot\alpha[/tex]

all these yield:
[tex]a=\frac{g(m_1+m_2)}{kM+m_1+m_2}[/tex]
While the answer should be:
[tex]a=\frac{g(m_1-m_2)}{kM+m_1+m_2}[/tex]
 

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  • #2
Karol said:

The Attempt at a Solution


[tex]m_1g-T_1=m_1a[/tex]
[tex]T_2-m_2=m_2a[/tex]
[tex]\mbox{ }R\alpha=a[/tex]
[tex](T_1-T_2)R=kMR^2\cdot\alpha[/tex]

All is correct up to here, but the missing g in the second equation.

Karol said:
all these yield:
[tex]a=\frac{g(m_1+m_2)}{kM+m_1+m_2}[/tex]

Check it.

T1=m1(g-a),
T2=m2(g+a)

T1-T2?ehild
 
  • #3
Thanks!
 

1. How does the mass of the pulley affect the system?

The mass of the pulley has no effect on the mechanics of the system. It only serves as a support for the masses and does not contribute to the overall motion or force.

2. What is the relationship between the two masses on the pulley?

The two masses on the pulley are connected by a rope or string that passes over the pulley. This creates an inverse relationship between the two masses, meaning that as one mass increases, the other will decrease and vice versa.

3. How does the friction of the pulley affect the motion of the masses?

The presence of friction on the pulley can affect the motion of the masses by reducing the amount of force transmitted through the rope or string. This can result in slower or more uneven motion of the masses.

4. What is the purpose of using a pulley in this system?

The pulley is used to redirect the force applied to the system and to change the direction of motion. It also allows for the use of multiple masses in the system, creating a more complex and realistic scenario for experimentation and analysis.

5. How does the acceleration of the masses change if the pulley is replaced with a frictionless one?

If the pulley is replaced with a frictionless one, the acceleration of the masses will increase. This is because without friction, there is no opposing force to slow down the motion of the masses, allowing them to accelerate more quickly.

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