A hypothetical force is described by the equation F(x) = -a/x^3 + bx^2, where a and b are positive constants. Find an expression for the amount of work that is done by this force over the range x = 1 to x = 3.
I am thinking to calculate the change in force and multiply it by change in distance...but what is change in distance?
work= INTEGRAL F(x)*dx from x=1 to 3
= INT (-a/x^3 + bx^2)dx over limits
= (a/2x^2 + bx^3/3) over limits
= a(1/18-1/2)+b(9-1/3) check that, I did it in my head.
To find the amount of work done by a force over a specific range, you need to calculate the integral of the force with respect to distance. In this case, the force is described by the equation F(x) = -a/x^3 + bx^2.
To calculate the change in distance, you need to subtract the initial position from the final position. In this case, the initial position is x = 1 and the final position is x = 3. Therefore, the change in distance is Δx = 3 - 1 = 2.
Now, let's find the expression for the amount of work done. The work done by a force is given by the integral of the force multiplied by the differential of the distance:
W = ∫ F(x) dx
Since the force varies with x, we need to integrate over the range from x = 1 to x = 3:
W = ∫[1,3] (-a/x^3 + bx^2) dx
To evaluate this integral, we can split it into two separate integrals:
W = ∫[1,3] (-a/x^3) dx + ∫[1,3] (bx^2) dx
The first integral, ∫(-a/x^3) dx, can be rewritten as -a ∫(x^(-3)) dx. Integrating this gives:
W = (-a) * (-(1/2) * x^(-2)) |[1,3] + ∫[1,3] (bx^2) dx
Simplifying further:
W = (a/2) * (1/x^2) |[1,3] + ∫[1,3] (bx^2) dx
Evaluating the expression at the upper and lower limits of integration:
W = (a/2) * ((1/3^2) - (1/1^2)) + ∫[1,3] (bx^2) dx
W = (a/2) * ((1/9) - 1) + ∫[1,3] (bx^2) dx
Now, integrate the second integral:
W = (a/2) * ((1/9) - 1) + (b/3) * (x^3) |[1,3]
W = (a/2) * ((1/9) - 1) + (b/3) * (3^3 - 1^3)
Simplifying further:
W = (a/2) * (-8/9) + (b/3) * (26)
Now, you can substitute the values of the constants a and b given in the problem to obtain the final expression for the amount of work done by the force over the range x = 1 to x = 3.