derivation for variable work energy theorem
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ork is a result of force and the resulting displacement. Now, we already know that all moving objects have kinetic energy. So, there must be a relation between Work and kinetic energy. This relation between the kinetic energy of an object and workdone is called “Work-Energy Theorem”. It is expressed as:
Work - Energy Theorem
Here, W is the work done in joules (J) and ΔK is the change in kinetic energy of the object. To learn how the Work-Energy Theorem is derived, we must first learn the nature of work as a scalar quantity and how two or more vector quantities are multiplied.
Proof of Work-Energy Theorem
We will look at the Work-Energy Theorem in two scenarios:
Workdone Under a Constant Force
We have already learnt about the equations of motion earlier and know that,
Work - Energy TheoremHere, v is the final velocity of the object; u is the initial velocity of the object; a is the constant acceleration and s is the distance traversed by the object. We can also write this equation as,
Work - Energy Theorem
We can substitute the values in the equation with the vector quantities, therefore:Work - Energy Theorem
If we multiply both sides with m/2, we get:
Work - Energy Theorem
From Newton’s second law, we know that F=ma, hence:
Work - Energy Theorem
Now, we already know that W= F.d and, K.E. = (mv²)/2,
So, the above equation may be rewritten as:
Work - Energy Theorem
Hence, we have:
Work - Energy Theorem
Therefore, we have proved the Work-Energy Theorem. The Work done on an object is equal to the change in its kinetic energy.
Workdone Under a Variable Force
A constant force is rare in the everyday world. A variable force is what we encounter in our daily life. Proving the Work-Energy Theorem for a variable force is a little tricky. If we take one axis as the Force applied and the other axis as displacement, we come up with the following graph:Work - Energy Theorem Here, x is the displacement. Now let’s divide this area into rectangles of infinitely small width along the x-axis:Work - Energy Theorem
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nee yeppo varuvennu sollu...
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