Show that work done is equal to dot product of force and displacement
Answers
Explanation:
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Answer:
Hello,
Work is said to be done whenever there is a transfer of energy. Mathematically, it is the dot product of force and displacement.
Suppose we have two line AB and CD and θis the angle between them. The component of AB long CD is given by ABcos θ. This is what we get when we do dot product. A force can do some work only when it has a component along the direction of the displacement. Hence, we take dot product of the force and displacement to find work done.
Explanation:
Work done is the product of the magnitude of the force acting and the displacement in the direction of the force.
If the two are all aligned we can simply say Work done = force x distance moved by force (WD = Fxd)
To cover the situation where the two are not perfectly aligned we can then say, well how much of that dispalcement is in the direction of the force? The component will be ‘d cos(theta)’. So work done = F x d cos(theta).
I am wondering whether the question is really asking about why the vector notation works. Maths is all made up stuff. People invented vectors. People invented (defined) operations between vectors and tended to invent ones which would probve to be useful to them.
They defined vectors and the dot product such that the vectors themselves match the properties of forces and displacements and that the dot product of two vectors a and b is equal to ab cos (theta). This means that the use of the vector description gives the same answers as the non- vector treatment.
An interesting issue is that sometimes maths was invented to describe physical situations ( Newton and calculus comes to mind) but at other times the maths existed anyway and was found to describe the physical situation. I believe calculus was separately invented by someone else( -Liebnitz?) and I dont know if this was in response to a need from a physics problem or more just a pure maths matter.
I wonder whether the question is really concerned aboout the fundamentals of ‘why work is equal to the dot product’ when I think the answer is that thay are just numerically equal