A potential energy function for a two-dimensional force is of the form U = 2.46x3 y - 5.05x (which has units of J). Calculate the force that acts at the point (1.26 m,1.18 m).
If
U(x,y) = 2.46x³y - 5.05x
what is U(1.26,1.18)?
To calculate the force that acts at a specific point given the potential energy function, we need to find the negative gradient of the potential energy function. The gradient of a scalar function in two dimensions gives us the direction and magnitude of the force acting at a point.
Let's start by finding the partial derivatives of the potential energy function U(x, y) with respect to x and y:
∂U/∂x = 3(2.46x^2)y - 5.05
∂U/∂y = 2.46x^3
Now, let's evaluate these partial derivatives at the point (1.26 m, 1.18 m):
∂U/∂x = 3(2.46(1.26)^2)(1.18) - 5.05 ≈ 8.7879 J/m
∂U/∂y = 2.46(1.26)^3 ≈ 4.053 J/m
Finally, we can find the force vector F(x, y) by combining these partial derivatives:
F(x, y) = (-∂U/∂x)i - (∂U/∂y)j
= (-8.7879 J/m)i - (4.053 J/m)j
The force vector acting at the point (1.26 m, 1.18 m) is approximately -8.7879 J/m in the x-direction and -4.053 J/m in the y-direction.