1.A body of mass m is slowly hauled up the hill by a force f which at each point was directed along a tangent to the trajectory .Find the work performed by this force , if the height of the hill is h and length of its base is L and coefficient of friction is k.Trajectory has not definite shape.
2.A body of mass m is hauled from earth's surface by applying a force F varying with height of ascent y as F = 2(ay - 1)mg, where a is a positive constant .Find the work performed by this force and the increment of the body's potential energy in the gravitational field of the Earth over the first half of the ascent.
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3 Answers
3411) Denote the path as P.
The work done by the non-conservative forces is equal to the change in mechanical energy which here is just mgh.(there is no change in KE)
Let the work done in pulling the block be W.
Then W - \int_P f_r dl = mgh
But f_r = \mu mg \cos \theta where \theta is the angle made by the tangent at a point on the path with the horizontal
\therefore \ \int_L f_r dl = \int_L \mu mg \cos \theta dl = \int_L \mu mg (dl \cos \theta)
But dl \cos \theta = dx where dx is measured along the horizontal (that is the projection of the displacement along the path, taken on the horizontal)
Now its obvious that \int_L dl \cos \theta = \int dx = L
Hence work done by friction = \mu mgL
W = \mu mgL+mgH which is independent of the path up the hill.
23use work done by force F + work done by gravity = change in KE
WF=∫F.dy
Wg=∫mg.dy