A thin uniform rod of length L pivots about its center. Two small thrusters attached at distances L/4 and L/2 from the pivot apply forces perpendicular to the rod in opposite directions, producing torques of equal magnitude τ0 but opposite sign. The thrusters fire according to the schedules:
• Thruster A (positive torque): τA(t)=+τ0(1−Tt) for 0≤t≤T, zero afterward.
• Thruster B (negative torque): τB(t)=−τ0(Tt) for 0≤t≤T, zero afterward.
(1) Part 1: (a) Write the algebraic expression for the net torque τnet(t) acting on the rod during 0≤t≤T.
(2) Part 2: (b) Derive an expression for the angular displacement Δθ of the rod during the firing interval in terms of τ0, T, and the rod’s moment of inertia I. Assume the rod is initially at rest in the horizontal plane.
(3) Part 3: (c) Using your result from part (b), determine the total work done on the rod by both thrusters during 0≤t≤T. Express your answer solely in symbols.
(4) Part 4: (d) Evaluate whether any additional work is done on the rod after t=T and justify your answer based on the physical situation.