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Consider a chain of mass $m$ whose one end is lifted upwards to a height $h$. The work done in this case would be $$\frac{mgh}{2}$$ But I didn't quite get why $mgh$ is divided by $2$. When I looked this up on the internet, it said that when objects like ladders, chains et cetera are lifted from one end, to calculate work done, the mass is divided by '2'. Still I didn't understand why. Please explain this to me (I don't know much about center of mass so please take that into consideration while answering).

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2 Answers 2

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The first thing we need to understand is the concept of center of mass. Newtonian mechanics is based on a point of an object. However, a chain, ladder, or most everyday objects are not a point in space, but a a collection of points in 3 dimension. Hence we might ask, how do we apply Newtonian mechanics to such objects. The answer, is to find a single point to represent the object, which is the center of mass. For the calcualtion of the position of center of mass you can refer to this link, but for chains, assuming it is a single line, the center of mass would be at the middle of the chain.

Second thing to consider, what work is. Work is defined as the force you apply to the object, multiplied with the distance travelled. Here, we know the force we need to apply is the same as the weight of the chain and we need to lift the end of a chain by $h$. However, we only need to consider the center of mass instead of the whole chain, which is at the center of the chain! Hence, the height considered is only half of $h$! It is not the mass that is divided by 2, but the height. Hence, we can then apply the force and the height into the formula and get the result of $W=F*d=(mg)*h/2$.

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When you are raising a chain (or a ladder) which has length $h$ then the top of the chain is raised through a height $h$ but the bottom of the chain is not raised at all, since it stays on the ground. Parts of the chain between the top and the bottom are raised by different heights. But it is intuitively clear that the average height through which the different parts of the chain are raised is $\frac h 2$.

Suppose a small part of the chain has mass $\delta m$ and so has weight $(\delta m)g$. As long as all parts of the chain are identical (so each unit length of the chain has the same mass) then the work done in raising the chain is the sum of $(\delta m)g$ times the average height through which the parts of the chain are raised. So

$\displaystyle W = \sum (\delta m) g \frac h 2 = \frac {mgh} 2$

This is an instance of the following general principle. As long as we considering linear motion (so the object is not rotating) then an extended object like a chain or a ladder can be treated as if its mass is concentrated at one point, called its centre of mass. If the chain or ladder has the same mass per unit length along its length then its centre of mass is at its middle. So if the chain or ladder has length $h$ ans its top is raised to a height $h$ while its other end stay on the ground then the work done is the same as raising a point particle with weight $mg$ by a distance $\frac h 2$.

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