Work-energy theorem from equation
dW=mv dv
Work done on the body in order to increase its velocity from u to v is given by
W=∫_u^v▒〖mv dv〗=m∫_u^v▒〖v dv〗=m[v^2/2]_u^v
W=1/2 m[v^2-u^2 ]
Work done = change in kinetic energy
W=∆E
This is work energy theorem, it states that work done by a force acting on a body is equal to the change in the kinetic energy of the body.
This theorem is valid for a system in presence of all types of forces (external or internal, conservative or non-conservative).
If kinetic energy of the body increases, work is positive i.e. body moves in the direction of the force (or field) and if kinetic energy decreases, work will be negative and object will move opposite to the force (or field).
Examples
i] In case of vertical motion of body under gravity when the body is projected up, force of gravity is opposite to motion and so kinetic energy of the body decreases and when it falls down, force of gravity is in the direction of motion so kinetic energy increases.
ii] When a body moves on a rough horizontal surface, as force of friction acts opposite to motion, kinetic energy will decrease and the decrease in kinetic energy is equal to the work done against friction.
THE WORK ENERGY THEOREM FOR VARIABLE FORCE
We are now familiar with the concepts of work and kinetic energy to prove the work-energy theorem for a variable force. We confine ourselves to one dimension. The time rate of change of kinetic energy is
dk/dt=d/dt (1/2 mv^2 )=m dv/dt v
= Fv (from Newton’s second Law)
=F dx/dt
Thus dk = Fdx
Integrating from the initial position (xi) to final position (xf), we have
∫_(k_1)^(k_1)▒dk=∫_(x_1)^(x_1)▒Fdx
where, Ki and Kf are the initial and final kinetic energies corresponding to xi and xf or Kf – Ki= W. Thus, the WE theorem is proved for a variable force.
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