Explain work energy principle using calculus method
Answers
Kinetic Energy: A force does work on the block. The kinetic energy of the block increases as a result by the amount of work. This relationship is generalized in the work-energy theorem.
The work W done by the net force on a particle equals the change in the particle’s kinetic energy KE:
W
=
Δ
KE
=
1
2.
mv
2
f
−
1
2
mv
2
i
where vi and vf are the speeds of the particle before and after the application of force, and m is the particle’s mass.
Derivation
For the sake of simplicity, we will consider the case in which the resultant force F is constant in both magnitude and direction and is parallel to the velocity of the particle. The particle is moving with constant acceleration a along a straight line. The relationship between the net force and the acceleration is given by the equation F = ma (Newton’s second law), and the particle’s displacement d, can be determined from the equation:
v
2
f
=
v
2
i
+
2
ad
obtaining,
d
=
v
2
f
−
v
2
i
2
a
The work of the net force is calculated as the product of its magnitude (F=ma) and the particle’s displacement. Substituting the above equations yields:
W
=
Fd
=
ma
v
2
f
−
v
2
i
2
a
=
1
2
mv
2
f
−
1
2
mv
2
i
=
KE
f
−
KE
i
=
Δ
KE