Question #ffb6c

2 Answers
May 7, 2017

#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#

May 7, 2017

#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#