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A block of mass 30kg is being brought down by a chain. If the block acquires a speed of \[40\,cm\,{{s}^{-1}}\] in dropping down \[2\,m\], find the work down by the chain during the process.\[(g=10m{{s}^{-2}})\]
(A). \[-297.6J\]
(B). \[-375J\]
(C). \[-597.6J\]
(D). \[-674.6J\]

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Last updated date: 14th Jul 2024
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Answer
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Hint: When the block is being brought down by the chain; two forces are acting on it. The force applied by the chain and the gravitational attraction are acting in opposite directions therefore both forces are doing some work on the block. The work done by the chain is equal to the difference in total work done on the block and work done by gravity.
Formulas Used:
\[E=W=\dfrac{1}{2}m{{v}^{2}}\]
\[\text{The work is done by the chain on the block} = \text{Total energy}- \text{Work done by gravity} \]

Complete step-by-step solution
A block of mass \[30\,kg\] held by a chain is moving down with velocity \[40\,cm\,{{s}^{-1}}=0.4m{{s}^{-1}}\]
Therefore the total work done on the block is –
\[E=W=\dfrac{1}{2}m{{v}^{2}}\]
Here, \[m\] is the mass of the block
\[v\] is the velocity of the block
 Substituting values in the above equation, we get,
\[W=\dfrac{1}{2}\times 30\times {{(0.4)}^{2}}\]
\[\Rightarrow W=2.4J\] - (1)
Therefore the total work done on the block is \[2.4J\]
The work done on the block by the gravity is given by-
\[{{W}_{g}}=mgh\]
Here, \[g\] is acceleration due to gravity
\[h\] is the height through which it was brought down
Therefore,
\[\begin{align}
  & {{W}_{g}}=30kg\times 10ms{}^{-2}\times 2m \\
 & {{W}_{g}}=600J \\
\end{align}\]
\[\text{The work is done by the chain on the block} = \text{Total energy}- \text{Work done by gravity} \]
\[\begin{align}
  & {{W}_{c}}=W-{{W}_{g}} \\
 & {{W}_{c}}=2.4-600 \\
 & {{W}_{c}}=-597.6J \\
\end{align}\]
The work done by the chain is \[-597.6J\], therefore the correct option is (C).

Note: When work done is negative, the work is being done on the system. When work is done is positive, the work is being done by the system. Work done by gravity is defined as the work done to take an object up to a height \[h\]. When the block is coming down, its potential energy is converted to kinetic energy.