Work done in bringing down a body A block of mass $30$ kg is being brought down by a chain. If the block acquires a speed of 40.0 cm/s in dropping down $2.00$ m, find the work done by the chain during the process.
I have calculated this. The acceleration of the block if the frame of reference is attached to the block, is $a-g=-9.76 $
So the work done comes out to be $-586$ J. 
What is the significance -ve sign here? And will the work done be +ve?
Also, the work done is also equal to the change in kinetic energy. Change in kinetic energy = $2.4$ J. This does not match the answer. Where am I wrong?
 A: What is the significance −ve sign here? And will the work done be +ve? 
If I want to lift some mass upwards I need to apply a force at least equal to it's weight. This means if I want to pull the mass upwards, with a certain acceleration then an additional force has to be supplied along with it's weight. But the resultant force will be the difference in these forces in magnitude. Now, to speak about the direction; w.r.t. (earth whose force acts downwards), the body is moving upwards with a force that can be defined from it's acceleration upwards. Hence the force defined by earth is negative. Now, according to me, it is pulled upward with some force that I exerted. So here the force is positive since the acceleration is in the same sense as the applied force. So the work done by the earth is negative while the work done by me is positive. The magnitudes of both coincide.   
The reverse is your case. Now you can think about it.  
Also, the work done is also equal to the change in kinetic energy. Change in kinetic energy =$2.4J$. This does not match the answer. 
According to work energy theorem, if the system is conservative, then the work done on a system will appear as the increase in kinetic energy of the system. But you have applied the wrong velocity in kinetic energy equation. There are something to be added here. The acceleration of the mass is not given. How you calculated that is also not provided. If your answer about the acceleration is right, read the remaining statements. The work done is calculated about a distance of $2m$ with an acceleration of $-9.76m/s^2$. So the velocity is not constant. You have to calculate the change in velocity over that distance. Apply that velocity and you will get the answer, which is the increase in kinetic energy of the mass throughout that distance.
(Hint: The acceleration is uniform over that distance. The final velocity is $40cm/s$. The initial velocity is to be calculate from Torricelli's equation. Find the change in velocity. Apply it in K.E. equation and you will get the answer.)
