To find the force exerted by the pulley on the objects, we start with the given values for tension ([tex]\( T(m) \)[/tex]) and the force of gravity ([tex]\( G(m) \)[/tex]). We need to calculate the negative sum of these values for various masses ([tex]\( m \)[/tex]). The force of the rope is defined as:
[tex]\[ \text{Force of the rope} = - (T(m) + G(m)) \][/tex]
Let's apply this formula step-by-step for each mass:
1. For [tex]\( m = 0.5 \)[/tex]:
[tex]\[ T(0.5) = 1.25 \][/tex]
[tex]\[ G(0.5) = 4.9 \][/tex]
[tex]\[ \text{Force of the rope} = - (1.25 + 4.9) = - (6.15) = -6.15 \][/tex]
2. For [tex]\( m = 1.2 \)[/tex]:
[tex]\[ T(1.2) = 3 \][/tex]
[tex]\[ G(1.2) = 11.76 \][/tex]
[tex]\[ \text{Force of the rope} = - (3 + 11.76) = - (14.76) = -14.76 \][/tex]
3. For [tex]\( m = 2.6 \)[/tex]:
[tex]\[ T(2.6) = 6.5 \][/tex]
[tex]\[ G(2.6) = 25.48 \][/tex]
[tex]\[ \text{Force of the rope} = - (6.5 + 25.48) = - (31.98) = -31.98 \][/tex]
4. For [tex]\( m = 3.4 \)[/tex]:
[tex]\[ T(3.4) = 8.5 \][/tex]
[tex]\[ G(3.4) = 33.32 \][/tex]
[tex]\[ \text{Force of the rope} = - (8.5 + 33.32) = - (41.82) = -41.82 \][/tex]
Combining these results into a table, we get:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$m$ & 0.5 & 1.2 & 2.6 & 3.4 \\
\hline
$-(T(m)+G(m))$ & -6.15 & -14.76 & -31.98 & -41.82 \\
\hline
\end{tabular}
\][/tex]
Therefore, the table that correctly displays the combined functions to find the force of the rope is the second table:
[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
$m$ & 0.5 & 1.2 & 2.6 & 3.4 \\
\hline
$-(T(m)+G(m))$ & -6.15 & -14.76 & -31.98 & -41.82 \\
\hline
\end{tabular}
\][/tex]
