Question

A 10-kilogram body is constrained to move along the x-axis. The potential energy U of the body in joules is given as a function of its position x in meters by U(x) = 6x² - 4x + 3 The force on the particle at x = 3 meters is (A) 32 N in +x direction (B) 32 N in - x direction (C) 45 N in +x direction (D) 45 N in - x direction (E) 98 N in + x direction

Answer 1

The force on a 10-kilogram body moving along the x-axis with the potential energy function U(x) = 6x² - 4x + 3 at x = 3 meters is found by differentiating the potential energy function and evaluating it at that point, resulting in a force of 32 N in the -x direction.

Step-by-step explanation:

The student's question pertains to the force experienced by a 10-kilogram body tied to a given potential energy function along the x-axis. According to physics and the concepts of mechanics, the force acting on a particle at a particular position in a potential field is the negative gradient of the potential energy function with respect to position. For a one-dimensional case along the x-axis, this translates to F(x) = -dU/dx, where U(x) is the potential energy given as a function of x.

In the given potential energy function U(x) = 6x² - 4x + 3, to find the force at x = 3 meters, we must take the derivative of U with respect to x and then insert the value x =3 m into the resulting expression. Here's how to calculate it step by step:

1. Calculate the derivative of U(x), F(x) = -dU/dx = -d/dx(6x² - 4x + 3).
2. The derivative of U(x) with respect to x is F(x) = -dU/dx = - (12x - 4).
3. Substitute x = 3 m into the derived force equation: F(3) = - (12(3) - 4) = - (36 - 4) = -32 N.

Therefore, the force on the particle at x = 3 meters is 32 N in the -x direction (option B).