A potential energy function for a two-dimensional force is of the form U = 3 x2y - 6 x. Find the force that acts at the point (x, y). (Use x and y as appropriate.)
I would do it as a vector..
Force=dU/di,j,k = (6xy-6)i + 3x^2 j
where i, j are unit x,y vectors.
To find the force that acts at a specific point (x, y) from a potential energy function, we need to calculate the negative gradient of the potential energy function.
The negative gradient of a scalar function is a vector that points in the direction of steepest descent and its magnitude represents the rate of change of the function in that direction.
Let's find the partial derivatives of the potential energy function U with respect to x and y:
∂U/∂x = 6xy - 6
∂U/∂y = 3x^2
Now, we can find the force by taking the negative gradient vector ∇U = (∂U/∂x, ∂U/∂y) and multiplying it by -1:
F = -∇U = (-∂U/∂x, -∂U/∂y)
F = (-6xy + 6, -3x^2)
Therefore, the force that acts at the point (x, y) is given by F = (-6xy + 6, -3x^2).