Please explain the given formula for acceleration and how it derived??

Question

I have doubt in the following formula:
#a=((m_1-m_2)*g )/((m_1+m_2))#
It is for the system of pulleys.
I have doubt in understanding how the formula is derived and In which situations can we use it.
Is there any other alternative for this formula?

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Answer 1 · 2017-07-31T12:42:48

See below.

Considering a pulley with an inextensible rope with two weights attached at both ends, for each weight we have

#T-m_1g = m_1 a_1#
#T-m_2g=m_2a_2#

but due to the inextensible rope

#a_2=-a_1 = -a# then

#T-m_1g = m_1 a#
#T-m_2g=-m_2a#

Here

#T = # rope tension.
#a_i =# weight accelerations.
#m_i =# weight masses.
#g = #gravity acceleration.

now eliminating the common rope tension #T# we have

#m_1a+m_1g = m_2g-m_2a# and then

#a = ((m_2-m_1)/(m_1+m_2))g#

If we consider instead that #a_2 = -a_1= a# then

#a = ((m_1-m_2)/(m_1+m_2))g#