We have to forces acting on the system (elevator+passengers):
1) The weight (W=5000 N), acting downward
2) The cable's tension (T=6000 N), acting upward
So, the two forces have opposite direction. The resultant (in upward direction) will be
![F=T-W](https://tex.z-dn.net/?f=F%3DT-W)
And for Newton's second law, the resultant of the forces acting on the system causes an acceleration on the system itself, given by
![a= \frac{F}{m}](https://tex.z-dn.net/?f=a%3D%20%5Cfrac%7BF%7D%7Bm%7D%20)
where m is the mass of the system.
So, we need to find F and m.
The resultant of the forces is
![F=T-W=6000 N-5000 N=1000 N](https://tex.z-dn.net/?f=F%3DT-W%3D6000%20N-5000%20N%3D1000%20N)
To find m, we can use the weight of the system. In fact, the weight of an object is given by
![W=mg](https://tex.z-dn.net/?f=W%3Dmg)
where
![g=9.81 m/s^2](https://tex.z-dn.net/?f=g%3D9.81%20m%2Fs%5E2)
. Solving for m, and using W=5000 N, we find
![m= \frac{W}{g}= \frac{5000 N}{9.81 m/s^2}=510 kg](https://tex.z-dn.net/?f=m%3D%20%5Cfrac%7BW%7D%7Bg%7D%3D%20%5Cfrac%7B5000%20N%7D%7B9.81%20m%2Fs%5E2%7D%3D510%20kg%20%20)
and at this point, we can calculate the acceleration of the system (elevator+people):
![a= \frac{F}{m}= \frac{1000 N}{510 kg}=1.96 m/s^2](https://tex.z-dn.net/?f=a%3D%20%5Cfrac%7BF%7D%7Bm%7D%3D%20%5Cfrac%7B1000%20N%7D%7B510%20kg%7D%3D1.96%20m%2Fs%5E2%20%20)
and the acceleration has the same direction of the resultant force, so upward.