Calculating Masses in a Dynamic Pulley System

[PhoenixWright]

Homework Statement

You have a a mass, m1, in a plane. This mass is connected to a pulley by a thread, to a mass m2. Prove that:

\begin{equation}
m_{1} = \dfrac{m_{2}(2g - a)}{4a}
\end{equation}

Homework Equations

The Attempt at a Solution

I don't know why, but I can get this:

\begin{equation}
m_{1} = \dfrac{m_{2}(g - a)}{a}
\end{equation}

It's a similar equation, but it's not the same. How can I get the right solution?

Thank you!

 

[drvrm]

I don't know why, but I can get this:

m1=m2(g−a)a​

(2)(2)m1=m2(g−a)a\begin{equation} m_{1} = \dfrac{m_{2}(g - a)}{a} \end{equation}

It's a similar equation, but it's not the same. How can I get the right solution?

how you take a start-give free body diagram of the masses?

 

[PhoenixWright]

how you take a start-give free body diagram of the masses?

m1 is in a plane (it's not inclinated) and m2 is is hanging from a pulley. They are connected by a thread.

In m1 we only consider Tension force (because N = mg), and in m2 we consider Tension force and m_{2}g

Thanks

 

[drvrm]

m1 is in a plane (it's not inclinated) and m2 is is hanging from a pulley. They are connected by a thread.

In m1 we only consider Tension force (because N = mg), and in m2 we consider Tension force and m_{2}g

Thanks

so , how you write the equation of motion? the pulley is massless or not? if m1 is sitting on a plane will there be motion?

 

[PhoenixWright]

so , how you write the equation of motion? the pulley is massless or not? if m1 is sitting on a plane will there be motion?

The pulley is massless. The plane has no motion.

I use this:
\begin{equation}
m_{1}\vec{g} + \vec{N} + \vec{T_{1}} = m_{1}\vec{a_{1}}
\ \\
m_{2}\vec{g} + \vec{T_{2}} = m_{2}\vec{a_{2}}
\end{equation}
Therefore:
\begin{equation}
T_{1} = m_{1}a_{1} \ \\
T_{2} - m_{2}g = m_{2}a_{2}
\ \\
T_{1} = T_{2} (massless)
\ \\
a_{1} = -a_{2}
\end{equation}
Then, I got the equation I put before... But it's not what I must get.

 

[drvrm]

The pulley is massless. The plane has no motion.

i just feel that how the mass m1 can move sitting tight on the plane- if it is raised from the plane by the string then what happens to gravitational pull-sorry i can not visualize it ,pl make the dynamics more clear

 

[PhoenixWright]

i just feel that how the mass m1 can move sitting tight on the plane- if it is raised from the plane by the string then what happens to gravitational pull-sorry i can not visualize it ,pl make the dynamics more clear

This is a pic of the situation:

 

[drvrm]

This is a pic of the situation:

in the above case your equations seem to be correct but the accelerations are not related like a(1) = -a(2) as one is horizontal and the other is vertical

 

[PhoenixWright]

in the above case your equations seem to be correct but the accelerations are not related like a(1) = -a(2) as one is horizontal and the other is vertical

Thanks, but, anyway, I can't get the solution if I suppose a1 = a2...

 

[PhoenixWright]

Problem solved. The exercise was wrong, there were 2 pulleys, not only 1.