A particle moves along the axis under the influence of a variable force = 6.8 + 4.4 where the force is measured in Newtons and the distance in meters. What is the potential energy associated with this force at = 4.0 m? Assume that = 0 J at = 0 m
Is there supposed to be a sqrt(-1) or an x after the 4.4? 6.8 + 4.4 is just 13.2. Is this one-dimensional motion? If so, just integrate -F dx for the potential energy.
To find the potential energy associated with a force, we need to integrate the force with respect to the distance.
In this case, the force F(x) = 6.8x + 4.4x^2, where x is the distance along the axis.
The potential energy U(x) associated with this force can be found by integrating F(x) with respect to x:
U(x) = ∫[6.8x + 4.4x^2] dx
To integrate the force, we can use the power rule of integration:
∫(ax^n) dx = (a/(n+1)) * x^(n+1) + C
Applying the power rule to each term of the force equation, we get:
U(x) = (6.8/2) * x^2 + (4.4/3) * x^3 + C
Since the potential energy U(x) is relative and can have an arbitrary constant C, we are only interested in the potential energy difference between two positions.
Now, to find the potential energy at x = 4.0 m, we substitute x = 4.0 into the equation:
U(4.0) = (6.8/2) * 4.0^2 + (4.4/3) * 4.0^3 + C
Simplifying the equation gives the potential energy at x = 4.0 m.