Question

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

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?

Answer 1

See below.

Explanation:

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`