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Solved Problem on Dynamics

In the system of the figure, the body *A* slides on a horizontal surface without friction, dragged by body
*B* that moves downward. Bodies *A* and *B* are attached by a rope of negligible mass parallel to the
surface and passing through a frictionless pulley of negligible mass. The masses of *A* and *B* are respectively
32 kg and 8 kg. Find the acceleration of the system and the tension on the cord. Assume
*g* = 10 m/s^{2}.

Problem data

- mass of body
*A*:*m*_{A}= 32 kg; - mass of body
*B*:*m*_{B}= 8 kg; - free-fall acceleration:
*g*= 10 m/s^{2}.

We choose the acceleration in the direction in which the body *B* moves downward. Drawing a free-body diagram for
each block and using *Newton's Second Law*

\[ \bbox[#99CCFF,10px]
{\vec{F}=m\vec{a}}
\]

figure 1

Solution

Body*A*

Vertical direction

Body

Vertical direction

- \( \vec P_{\text{A}} \): weight of body
*A*; - \( \vec N_{\text{A}} \): normal force of the surface on the body.

- \( \vec T \): tension on the cord.

figure 2

In the vertical direction, the weight and the normal force cancel out, there is no vertical motion.

In the horizontal direction using*Newton's Second Law* we have

In the horizontal direction using

\[
T=m_{\text{A}}a \tag{I}
\]

Body *B*
*Newton's Second Law* we have

- \( \vec W_{\text{B}} \): weight of body
*B*; - \( \vec T \): tension on the rope.

figure 3

\[
W_{\text{B}}-T=m_{\text{B}}a \tag{II}
\]

Expressions (I) and (II) can be written as a system of linear equations with two variables (

\[
\frac{
\left\{
\begin{array}{rr}
\cancel{T}&=m_{\text{A}}a\\
W_{\text{B}}-\cancel{T}&=m_{\text{A}}a
\end{array}
\right.}
{W_{\text{B}}=\left(m_{\text{A}}+m_{\text{B}}\right)a}
\]

\[
a\;=\;\frac{W_{\text{B}}}{m_{\text{A}}+m_{\text{B}}} \tag{III}
\]

The weight of body
\[
W_{\text{B}}=m_{\text{B}}g \tag{IV}
\]

substituting (IV) into (III) and the values given in the problem
\[
a=\frac{m_{\text{B}}g}{m_{\text{A}}+m_{\text{B}}}\\
a=\frac{8 \times 10}{32+8}\\
a=\;\frac{80}{40}
\]

\[ \bbox[#FFCCCC,10px]
{a=2\;\text{m/s}^{2}}
\]

Substituting the mass of the body
\[
T=32 \times 2
\]

\[ \bbox[#FFCCCC,10px]
{T=64\ \text{N}}
\]

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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 .