Question

Question #ffb6c

Answer 1

`F = - (dU)/(dx) implies U = - int 6 x-12 dx = - 3x^2 + 12 x + C`

`U(0) = 20 implies C = 20`

`implies U(x) = 20 + 12x - 3x^2`

So:

`U(3) = 29 J`

Answer 2

`29J`

Explanation:

If a force which acts on an object is a function of position only, it is called a conservative force.
It can be represented by a potential energy function which for a one-dimensional case satisfies the following condition

`- (dU(x))/(dx)=F(x)`

or in integral form as

`U(x)=-int_(x_0)^xF(x) dx`
where `x_0` is the reference point

Inserting given function we get
`U (x)= - int( 6 x-12 )dx`
`=>U(x)=- 6x^2/2 + 12 x + C` .......(1)
where `C` is constant of integration.

To evaluate `C` we find the value of function `U(x)` at `x=0`. We have
`- 3xx0^2 + 12 xx0 + C=20`
`=>C = 20`
From (1) we get
`U(x) = - 3x^2+12x+20`

`:.U(3) = - 3xx3^2+12xx3+20=29 J`