This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Step 1: Write the formula for work done. The work done W by a force F along a curve C is given by the line integral: W = _C F · dr The force is given as F = 3x^2 i + (2x-y) j + z k. The curve is parameterized by x = 2t^2, y = t, z = 4t^2 - t, from t=0 to t=1. Step 2: Express F in terms of t. Substitute x = 2t^2, y = t, and z = 4t^2 - t into the expression for F: F_x = 3x^2 = 3(2t^2)^2 = 3(4t^4) = 12t^4 F_y = 2x-y = 2(2t^2) - t = 4t^2 - t F_z = z = 4t^2 - t So, F(t) = 12t^4 i + (4t^2 - t) j + (4t^2 - t) k. Step 3: Find dr in terms of t. The position vector is r(t) = x(t) i + y(t) j + z(t) k. r(t) = 2t^2 i + t j + (4t^2 - t) k Now, find the differential dr = drdt dt: (dx)/(dt) = (d)/(dt)(2t^2) = 4t (dy)/(dt) = (d)/(dt)(t) = 1 (dz)/(dt) = (d)/(dt)(4t^2 - t) = 8t - 1 So, dr = (4t i + 1 j + (8t - 1) k) dt. Step 4: Calculate the dot product F · dr. F · dr = (F_x (dx)/(dt) + F_y (dy)/(dt) + F_z (dz)/(dt)) dt F · dr = ( (12t^4)(4t) + (4t^2 - t)(1) + (4t^2 - t)(8t - 1) ) dt F · dr = ( 48t^5 + 4t^2 - t + (32t^3 - 4t^2 - 8t^2 + t) ) dt F · dr = ( 48t^5 + 4t^2 - t + 32t^3 - 12t^2 + t ) dt Combine like terms: F · dr = ( 48t^5 + 32t^3 + (4t^2 - 12t^2) + (-t + t) ) dt F · dr = (48t^5 + 32t^3 - 8t^2) dt Step 5: Integrate from t=0 to t=1. W = _0^1 (48t^5 + 32t^3 - 8t^2) dt W = [ (48t^6)/(6) + (32t^4)/(4) - (8t^3)/(3) ]_0^1 W = [ 8t^6 + 8t^4 - (8)/(3)t^3 ]_0^1 Step 6: Evaluate the definite integral. W = ( 8(1)^6 + 8(1)^4 - (8)/(3)(1)^3 ) - ( 8(0)^6 + 8(0)^4 - (8)/(3)(0)^3 ) W = ( 8 + 8 - (8)/(3) ) - (0) W = 16 - (8)/(3) To subtract, find a common denominator: W = (16 × 3)/(3) - (8)/(3) W = (48)/(3) - (8)/(3) W = (40)/(3) The work done is (40)/(3).