Solved Problem on Static Equilibrium
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A body with weight W is suspended by a system of pulleys and strings. Assuming these elements are
lightweight and the pulleys and ropes have no friction. Determine:
a) The force that man must apply on the rope to keep the body in static equilibrium;
b) If the rope is pulled down 60 cm, how much does the body lifts.
a) The force that man must apply on the rope to keep the body in static equilibrium;
b) If the rope is pulled down 60 cm, how much does the body lifts.
Problem data:
- Weight of the body: W.
a) To keep the body in static equilibrium, the force exerted by man should be equal to the gravitational force, of
the body, from the static equilibrium
\[
\begin{gather}
\vec{F}={\vec F}_{g} \tag{I}
\end{gather}
\]
In the body acts the gravitational force
\( {\vec F}_{g} \)
and tension force
\( \vec{T} \)
on the rope
\[
\begin{gather}
{\vec F}_{g}=\vec{T} \tag{II}
\end{gather}
\]
The tension force is transmitted through the rope to the pulley, where the body is attached. For the
equilibrium of the system the tension force should divide itself between the two sides of the pulley,
for each side, we have a tension equal to
\( \frac{\vec T}{2} \)
acting on the rope. The lowest pulley is fixed to another pulley where acts this tension
\( \frac{\vec T}{2} \),
for the equilibrium of the system the tension should also be divided between the two sides of the middle
pulley, for each side, we will have a tension equal to
\( \frac{\vec T}{4} \).
This tension is transmitted by the rope to the pulley fixed on the ceiling, this pulley only transmits
the tension from one side to another where the man holds the rope. From the expression (II), we have
that the gravitational force that will act on the rope at this point will be
\( \frac{{\vec F}_{g}}{4} \),
a system of two-pulley divides the weight of the load by 4. Then from the expression (I) the force that
man exerts
Figure 1
\[
\begin{gather}
\bbox[#FFCCCC,10px]
{F=\frac{F_{g}}{4}}
\end{gather}
\]
b) When the rope is pulled 60 cm down a point A on the rope (Figure 2) descends the 60 cm, as
this pulley is fixed on the ceiling, the rope on the other side of the pulley rises 60 cm. As the middle
pulley is free, a point B on the rope should rise the same 60 cm, of this value 30 cm is the
displacement of the point, and the other 30 cm are resulting from the pulley itself that is pulled up.
As the middle pulley rose 30 cm a point C on the rope of the pulley underneath should rise 30 cm,
the pulley underneath is free, then of this value, 15 cm will be the displacement of the point, and the
other 15 cm represent the rise of the pulley. As the body is attached to this pulley it will rise 15 cm
along with the pulley.
The body will be rise from 15 cm. 15 cm.
The body will be rise from 15 cm. 15 cm.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .