Work is said to be done when applied force is able to produce displacement. Work can be mathematically expressed as the dot product of force and displacement. Let us consider a uniform rod of length L and mass M on the horizontal surface of the ground. Now we want to lift that body from the ground in such a way that one end of the rod is going to stay there in the ground itself and the other end is lifted to a certain height or certain angle with the horizontal. For this to happen, we shall apply some force and the body gets displaced with respect to the applied force.
Uniform rod is a body but not a point mass. Here mass is uniformly distributed and all over the length. If we have to consider this body as a point mass where its mass is concentrated, we shall get concept of center of mass into consideration.
Center of mass is a point of a body where its entire mass appears to be concentrated and it the point of the body which represents the translatory motion of the body. For a regular body, it is at the geometric centre of the body. For the rod of uniform length, hence center of mass is at the half of its physical length.
We shall apply a minimum force that is equal to the weight of the body so that body can start moving. Though entire body is moved up to a certain height, we have to consider the point of center of mass to know up to where the body got lifted. The height can be expressed in terms of SIN angle and the length of the rod.
Depending on up to what angle we are lifting the rod, work done as also going to be different.
Further if we are lifting another body of certain mass by keeping it on the movable end, we need to do work to lift both of them. For the rod, the treatment is same, where as the for the point sized body, it is like the point of displacement is with in the body it self and we do not need the concept of center of mass.
The detailed mathematical treatment is done in the following video as shown below.
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