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Physics+ Work Lemmas: UNIZOR.COM - Classic Physics+ - Laws of Newton
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Field Work Lemmas
In the previous lecture we have introduced the concepts of a field and field intensity force that is equal to a gradient of the field potential.
Also, we have proven that, dealing with such force, the work of this field intensity force along any trajectory of an object moving in the field depends only on the field potential at the beginning and at the end of a trajectory and is independent of a path between these two points.
There is a converse theorem that states that, if the work performed by some force on an object depends on the object's position in the beginning and at the end of its movement and does not depend on a trajectory between these points, then this force can be represented as a gradient of some scalar function, the field potential.
This lecture presents certain auxiliary theorems (lemmas) that will help to prove the above mentioned theorem in the next lecture.
Lemma A
Compare the work W[AB] done by the force F along the object's trajectory from point A to point B with the work W[BA] done when an object moves from point B to point A along the same trajectory in the opposite direction.
Prove that W[AB] = −W[BA]
Lemma B
It's given that the work of this force on an object moving along any trajectory between any pair of points A and B depends only on a choice of these two endpoints and does not depend on a choice of trajectory between them.
Prove that the work this force performs on an object moving along a closed trajectory, when points A and B coincide, equals to zero.
Lemma C
It's given that the work of this force on an object moving along any closed trajectory that starts and ends at the same point equals to zero.
Prove that the work this force performs on an object moving from any fixed point A to any fixed point B does not depend on trajectory between these points.
Lemma D
Assume, there are three points in the area where the force is acting, A, B and C.
Consider amounts of work the force performs on an object during its movements between these points W[AB], W[AC], W[BC].
Prove that W[BC] = W[AC] − W[AB]
Read full text of notes for this lecture on UNIZOR.COM - Classic Physics+ - Laws of Newton - Work Lemmas
Notes to a video lecture on UNIZOR.COM
Field Work Lemmas
In the previous lecture we have introduced the concepts of a field and field intensity force that is equal to a gradient of the field potential.
Also, we have proven that, dealing with such force, the work of this field intensity force along any trajectory of an object moving in the field depends only on the field potential at the beginning and at the end of a trajectory and is independent of a path between these two points.
There is a converse theorem that states that, if the work performed by some force on an object depends on the object's position in the beginning and at the end of its movement and does not depend on a trajectory between these points, then this force can be represented as a gradient of some scalar function, the field potential.
This lecture presents certain auxiliary theorems (lemmas) that will help to prove the above mentioned theorem in the next lecture.
Lemma A
Compare the work W[AB] done by the force F along the object's trajectory from point A to point B with the work W[BA] done when an object moves from point B to point A along the same trajectory in the opposite direction.
Prove that W[AB] = −W[BA]
Lemma B
It's given that the work of this force on an object moving along any trajectory between any pair of points A and B depends only on a choice of these two endpoints and does not depend on a choice of trajectory between them.
Prove that the work this force performs on an object moving along a closed trajectory, when points A and B coincide, equals to zero.
Lemma C
It's given that the work of this force on an object moving along any closed trajectory that starts and ends at the same point equals to zero.
Prove that the work this force performs on an object moving from any fixed point A to any fixed point B does not depend on trajectory between these points.
Lemma D
Assume, there are three points in the area where the force is acting, A, B and C.
Consider amounts of work the force performs on an object during its movements between these points W[AB], W[AC], W[BC].
Prove that W[BC] = W[AC] − W[AB]
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