The force on a particle is directed along an x axis and given by F = 1.7(x/2.8 - 0.95) where x is in meters and F is in Newtons. Find the work done by the force in moving the particle from x = 0 to x = 6.9 m.
To find the work done by a force in moving a particle from one point to another, you can use the formula:
Work = ∫(F(x) * dx)
where F(x) is the force as a function of position x, and dx represents an infinitesimally small displacement.
In this case, the force is given by F = 1.7(x/2.8 - 0.95), and we need to find the work done in moving the particle from x = 0 to x = 6.9 m.
To calculate the work, we first need to integrate the force function over the displacement range. Let's break it down step by step:
1. Integrate F(x) with respect to x to get the work function:
∫(F(x) * dx) = ∫[1.7(x/2.8 - 0.95)] * dx
2. Simplify the expression:
∫[1.7(x/2.8 - 0.95)] * dx = ∫[1.7x/2.8 - 1.7*0.95] * dx
= ∫(1.7x/2.8) * dx - 1.7 * ∫(0.95) * dx
= (1.7/2.8) ∫(x * dx) - 1.7 * (0.95) ∫(dx)
= (1.7/2.8)(x^2/2) - 1.7 * (0.95)(x) + C
= (0.60714)(x^2/2) - (1.615)(x) + C
3. Evaluate the work over the range x = 0 to x = 6.9 m:
Work = [(0.60714)(6.9^2/2) - (1.615)(6.9)] - [(0.60714)(0^2/2) - (1.615)(0)]
= [(0.60714)(47.61/2) - (1.615)(6.9)] - [(0.60714)(0/2) - (1.615)(0)]
= [(0.60714)(23.805) - (1.615)(6.9)]
= [14.442 - 11.1985]
= 3.2435 Joules
Therefore, the work done by the force in moving the particle from x = 0 to x = 6.9 m is approximately 3.2435 Joules.