Nozzle stream curvature; dependent motion

Nozzle stream

The stream leaves A at v = 25 m/s along a slope 3 : 4, so θ = tan⁻¹(3/4) = 36.87° (cos θ = 0.8, sin θ = 0.6). The only acceleration is g = 9.81 m/s² downward. Radius of curvature ρ = v²/a_n, where a_n is the component of g normal to the path.

(a) As it leaves the nozzle: a_n = g cos θ = 9.81 × 0.8 = 7.848 m/s² ρ_A = 25²/7.848 = 79.6 m

(b) At the maximum height B: velocity is horizontal: v_B = v_x = 25 × 0.8 = 20 m/s; g is entirely normal, so a_n = 9.81 m/s². ρ_B = 20²/9.81 = 40.8 m

(For reference, the maximum height is (25 × 0.6)²/(2 × 9.81) = 11.47 m above the nozzle and the time to reach it is 15/9.81 = 1.53 s.)

Dependent motion of particles

When the position of one particle depends on the position of another because they are joined by an inextensible cord (or a rigid link), their motions are called dependent. The constraint gives a relation between the coordinates; differentiating it once gives the velocity relation and twice the acceleration relation.

Example: block A moves on a horizontal table and is joined by a cord over a pulley to block B, which hangs from a movable pulley supported by the same cord. If x_A is the distance of A from the fixed pulley and y_B the drop of the movable pulley, the cord length is constant: x_A + 2y_B = L

so v_A = −2v_B and a_A = −2a_B. If B moves down 1 m, A moves 2 m toward the pulley. Only one coordinate is independent, so the system has one degree of freedom.