Question

Show the work energy theorem for a variable force.​

Answer 1

Now we can define the work done as a definite integral of the force over the total displacement as Now, we already know that kinetic energy of an object can be expressed as: We can express the change in the kinetic energy with time as, Hence, the work-energy theorem is proved for the variable force as well.

Answer 2

Answer:

Work energy theorem -

It states that the work done by the net force acting on a body is equal to the changed produced in kinetic energy of the body.

Let F be the variable force

∴ Work done by this variable force , W=∫

x

i

x

f

F⋅dx

where x

i

is the initial position

and x

f

is final position.

Also Kinetic energy of an object , K=

2

1

mv

2

⟹

dt

dK

=mv

dt

dv

⟹

dt

dK

=ma

dt

dx

⟹

dt

dK

F

dt

dx

⟹dK=Fdx

⟹∫

K

i

K

f

dK=∫

x

i

x

f

F⋅dx

⟹ΔK=W

Where ΔK is the change in kinetic energy