Question

The potential energy associated with a certain force is given by U(x) = -3x +4x^2, where x is in
meters and U is in joules. What is the value of the force when x = 1 m?

Answer 1

So then its 1J-1m=-3m+4m^2
Joule is eqal to kg*m^2/s^2

Answer 2

(kg*m^2/s^2 ) * m = -3m +4m^2
I thought I could divide everything by a meter...

Answer 3

which would give you kg*m^2/s^2 = -3 +4m
Now I'm stuck... help?

Answer 4

Am I even going about this the right way?

Answer 5

What is the relation between for energy level and force? How do you write an equation linking the two explicitly?

Answer 6

Remember that work/energy, U, is the integral of force F over the distance the force is applied:
\[ U = \int_A^B F \cdot d\ell \]
If we have a one-dimensional problem, then the dot product is just ordinary multiplication because the angle between F and dl is zero, hence
\[ U(B) - U(A) = \int_A^B F(x) \ dx \]

Now given that, if you know the function U(x), how do you find the function F(x)?

Answer 7

For example: gravity. There F(x) is a constant, F(g) = mg. If I raise an object from the ground at zero to a height h, the potential energy is
\[ U(h) - U(0) = \int_0^h mg \ dx = mgh \]

Answer 8

If your force is conservative then it can be written as the gradient of a potential:
\[\vec{F}=- \vec{\nabla}U\]
In one variable you get:
\[\vec{F}=-\frac{dU}{dx}\vec{e}_x\]

Answer 9

wouldnt the answer be 5??? or -5?

Answer 10

The derivative of
U(x) = -3x +4x^2,

is
dU/dx = -3 + 8x

hence
F = - dU/dx = 3 - 8x

and for x = 1,
F = 3 - 8 = -5