Answer:
![\mathbf{u(t) =\dfrac{m_2g }{k}(1 - cos \omega_n t) + \dfrac{\sqrt{2gh}}{\omega_n}\times \dfrac{m_1}{m_1+m_2}sin \omega _n t}](https://tex.z-dn.net/?f=%5Cmathbf%7Bu%28t%29%20%3D%5Cdfrac%7Bm_2g%20%7D%7Bk%7D%281%20-%20cos%20%5Comega_n%20t%29%20%2B%20%5Cdfrac%7B%5Csqrt%7B2gh%7D%7D%7B%5Comega_n%7D%5Ctimes%20%5Cdfrac%7Bm_1%7D%7Bm_1%2Bm_2%7Dsin%20%5Comega%20_n%20t%7D)
Explanation:
From the information given:
The equation of the motion can be represented as:
![(m_1 +m_2) \hat u + ku = m_2 g--- (1)](https://tex.z-dn.net/?f=%28m_1%20%2Bm_2%29%20%5Chat%20u%20%2B%20ku%20%3D%20m_2%20g---%20%281%29)
where:
= mass of the body 1
= mass of the body 2
= acceleration
k = spring constant
u = displacement
g = acceleration due to gravity
Recall that the formula for natural frequency ![\omega _n = \sqrt{\dfrac{k}{m_1+m_2}}](https://tex.z-dn.net/?f=%5Comega%20_n%20%3D%20%5Csqrt%7B%5Cdfrac%7Bk%7D%7Bm_1%2Bm_2%7D%7D)
And the equation for the general solution can be represented as:
![u(t) = A cos \omega_nt + B sin \omega _n + \dfrac{m_2g}{k} --- (2)](https://tex.z-dn.net/?f=u%28t%29%20%3D%20A%20cos%20%5Comega_nt%20%2B%20B%20sin%20%5Comega%20_n%20%2B%20%5Cdfrac%7Bm_2g%7D%7Bk%7D%20---%20%282%29)
To determine the initial velocity, we have:
![\hat u_2^2 = 2gh](https://tex.z-dn.net/?f=%5Chat%20u_2%5E2%20%3D%202gh)
![\hat u_2 = \sqrt{2gh}](https://tex.z-dn.net/?f=%5Chat%20u_2%20%3D%20%5Csqrt%7B2gh%7D)
where h = height
Suppose we differentiate equation (2) with respect to time t; we have the following illustration:
![\hat u (t) = - \omega_n A sin \omega_n t+ \omega_n B cos \omega _n t + 0](https://tex.z-dn.net/?f=%5Chat%20u%20%28t%29%20%3D%20-%20%5Comega_n%20A%20sin%20%5Comega_n%20t%2B%20%5Comega_n%20B%20cos%20%5Comega%20_n%20t%20%2B%200)
now if t = 0
Then
![\hat u (0) = - \omega_n A sin \omega_n (0)+ \omega_n B cos \omega _n (0) + 0](https://tex.z-dn.net/?f=%5Chat%20u%20%280%29%20%3D%20-%20%5Comega_n%20A%20sin%20%5Comega_n%20%280%29%2B%20%5Comega_n%20B%20cos%20%5Comega%20_n%20%280%29%20%2B%200)
![= \omega _n B](https://tex.z-dn.net/?f=%3D%20%5Comega%20_n%20B)
Using the law of conservation of momentum on the impact;
![m_2 \hat u_2=(m_1+m_2) \hat u (0)](https://tex.z-dn.net/?f=m_2%20%5Chat%20%20u_2%3D%28m_1%2Bm_2%29%20%5Chat%20u%20%280%29)
By replacing the value of
with ![\sqrt{2gh](https://tex.z-dn.net/?f=%5Csqrt%7B2gh)
Then the above equation becomes:
![m_2 \times \sqrt{2gh}=(m_1+m_2) \ u(0)](https://tex.z-dn.net/?f=m_2%20%5Ctimes%20%5Csqrt%7B2gh%7D%3D%28m_1%2Bm_2%29%20%5C%20u%280%29)
Making u(0) the subject of the formula, we have:
![u(0)= \dfrac{ m_2 \times \sqrt{2gh}}{(m_1+m_2)}](https://tex.z-dn.net/?f=u%280%29%3D%20%5Cdfrac%7B%20m_2%20%5Ctimes%20%5Csqrt%7B2gh%7D%7D%7B%28m_1%2Bm_2%29%7D)
Similarly, the value of the variable can be determined as follows;
Using boundary conditions
![0 = A cos 0 + B sin 0 + \dfrac{m_2g}{k}](https://tex.z-dn.net/?f=0%20%3D%20A%20cos%200%20%2B%20B%20sin%200%20%2B%20%5Cdfrac%7Bm_2g%7D%7Bk%7D)
![0 = A (1)+0+ \dfrac{m_2g}{k}](https://tex.z-dn.net/?f=0%20%3D%20A%20%281%29%2B0%2B%20%5Cdfrac%7Bm_2g%7D%7Bk%7D)
![A =- \dfrac{m_2g}{k}](https://tex.z-dn.net/?f=A%20%3D-%20%5Cdfrac%7Bm_2g%7D%7Bk%7D)
Also, if ![\hat u (0) = \omega_nB](https://tex.z-dn.net/?f=%5Chat%20u%20%280%29%20%3D%20%5Comega_nB)
Then :
![\dfrac{m_2}{m_1+m_2}\sqrt{2gh} = \omega_n B](https://tex.z-dn.net/?f=%5Cdfrac%7Bm_2%7D%7Bm_1%2Bm_2%7D%5Csqrt%7B2gh%7D%20%3D%20%5Comega_n%20B)
making B the subject; we have:
![B = \dfrac{m_2}{m_1 + m_2}\dfrac{\sqrt{2gh}}{\omega_n}](https://tex.z-dn.net/?f=B%20%3D%20%5Cdfrac%7Bm_2%7D%7Bm_1%20%2B%20m_2%7D%5Cdfrac%7B%5Csqrt%7B2gh%7D%7D%7B%5Comega_n%7D)
Finally, replacing the value of A and B back to the general solution at equation (2); we have the equation of the subsequent motion u(t) which is:
![\mathbf{u(t) =\dfrac{m_2g }{k}(1 - cos \omega_n t) + \dfrac{\sqrt{2gh}}{\omega_n}\times \dfrac{m_1}{m_1+m_2}sin \omega _n t}](https://tex.z-dn.net/?f=%5Cmathbf%7Bu%28t%29%20%3D%5Cdfrac%7Bm_2g%20%7D%7Bk%7D%281%20-%20cos%20%5Comega_n%20t%29%20%2B%20%5Cdfrac%7B%5Csqrt%7B2gh%7D%7D%7B%5Comega_n%7D%5Ctimes%20%5Cdfrac%7Bm_1%7D%7Bm_1%2Bm_2%7Dsin%20%5Comega%20_n%20t%7D)