Question

Two blocks of masses `m_1` and `m_2` are connected with a light spring of force constant `K` and whole system is kept on a frictionless horizontal surface the masses are applied forces `f_1` and `f_2` respectively?

At any time the blocks have accelerations a. The value of spring force is?

Answer 1

See below.

Explanation:

If the acceleration is the same then

`m_1 vec a = vec f_1 + vec T`
`m_2 vec a = vec f_2 - vec T`

adding both equations

`vec a = 1/(m_1+m_2)(vec f_1+vec f_2)`

and now subtracting

`vec T = 1/2(m_1-m_2)vec a -1/2(vec f_1-vec f_2) =`

`=1/2(m_1-m_2)/(m_1+m_2)(vec f_1+vec f_2)-1/2(vec f_1-vec f_2)`

`T = 1/2( (m_1 - m_2)/ (m_1 + m2)-1)vec f_1 + 1/2( (m_1 - m_2)/ (m_1 + m_2)+1)vec f_2`