Single Pulleys, Acceleration and Thrust
Acceleration
\[
F = ma
\]
\[
M_1 g - T = M_1 a
\]
\[
T - M_2 g = M_2 a
\]
\[
\text{adding}
\]
\[
M_1 g - M_2 g = M_1 a + M_2 a
\]
\[
M_1 g - M_2 g = a\,(M_1 + M_2)
\]
\[
\frac{M_1 g - M_2 g}{M_1 + M_2} = a
\]
\[
a = \frac{g(M_1 - M_2 )}{M_1 + M_2}
\]
Thrust
\[
T = M_1 g - M_1 a
\]
\[
T = M_2 a + M_2 g
\]
Adding the force equations:
\[
2T
= M_1 g - M_1 a + M_2 a + M_2 g
\]
\[
2T
= a\,(M_2 - M_1) + g\,(M_1 + M_2)
\]
Substituting the expression for \(a\):
\[
2T
=
\frac{g\,(M_1 - M_2)}{M_1 + M_2}\,(M_2 - M_1)
\;+\;
g\,(M_1 + M_2)
\]
Which simplifies to:
\[
T = \frac{2gM_1 M_2}{M_1 + M_2}
\]
