Final answer:
To calculate the force on a particle at a specific position x using the potential energy function U(x), differentiate U(x) concerning x, then substitute the given position. The result is a force of -5.112 Newtons at x = 0.38 m.
Step-by-step explanation:
To find the force on a particle given the potential energy function U(x) = A x² + B x + C, we use the concept that the force exerted on a particle at a position x is the negative derivative of the potential energy concerning position, F(x) = -dU/dx.
First, we find the derivative of the given potential energy function:
Then we plug in our constants and position (38 cm converted to meters, which is 0.38 m):
- F(x) at x = 0.38 m = -(2(1.2 J/m²)(0.38 m) + 4.2 J/m)
Calculating, we find:
- F(0.38) = -(2(1.2)(0.38) + 4.2)N = -(0.912 + 4.2)N = -5.112 N
Therefore, the force on the particle at x = 0.38 m is -5.112 Newtons.