As a particle moves along the x axis it is acted upon by a single conservative force given by Fx = (20 - 4,0x) N where x is in m. The potential energy associated with this force has the value +30 J at the origin 9x =0). What is the value of the potential energy at x = 4.0 m?
To find the value of the potential energy at x = 4.0 m, we can use the relationship between potential energy and conservative force.
The potential energy associated with a conservative force is determined by the negative of the work done by that force. The formula for calculating the work done by a force is given by:
Work = ∫ F(x) dx
In this case, the force F(x) = (20 - 4.0x) N, where x is in meters. To calculate the work done, we need to integrate this force equation over the interval from x = 0 to x = 4.
Work = ∫ (20 - 4.0x) dx
Integrating this expression gives us:
Work = [20x - 2.0x^2] evaluated from x = 0 to x = 4
Substituting the limits of integration:
Work = [20(4) - 2.0(4^2)] - [20(0) - 2.0(0^2)]
Work = [80 - 32] - [0 - 0]
Work = 48 J
Since potential energy is the negative of the work done, the potential energy at x = 4.0 m is -48 J.
However, the question states that the potential energy at the origin (x = 0) is +30 J. This means that we need to add the value of the potential energy at the origin to our calculated result:
Potential energy at x = 4.0 m = -48 J + 30 J
Potential energy at x = 4.0 m = -18 J
Therefore, the value of the potential energy at x = 4.0 m is -18 J.