Dependent motion; radius of curvature of particle path

Dependent motion of particles

The motion of one particle is dependent on that of another when the two are connected by an inextensible cord (or rigid link) so that their positions are related by a constraint equation.

Example: block A on a horizontal table is tied by a cord that passes over a fixed pulley to a block B hanging below a movable pulley. The cord length is constant: x_A + 2y_B = L (x_A measured from the fixed pulley, y_B downward). Differentiating: v_A = −2v_B and a_A = −2a_B, so if B goes down at 1 m/s, A moves at 2 m/s toward the pulley. One coordinate fixes the other.

Curved motion with constant acceleration

Data: a_x = −2 m/s², a_y = −5 m/s², u = 60 m/s at 30°: u_x = 60 cos 30° = 51.96 m/s, u_y = 60 sin 30° = 30 m/s. At t = 3 s:

• v_x = u_x + a_x t = 51.96 − 2(3) = 45.96 m/s
• v_y = u_y + a_y t = 30 − 5(3) = 15.00 m/s
• Speed v = √(45.96² + 15²) = 48.35 m/s, direction 18.1° above the x-axis.
• Total acceleration a = √(2² + 5²) = 5.385 m/s².
• Tangential component: a_t = (a_x v_x + a_y v_y)/v = (−2 × 45.96 − 5 × 15)/48.35 = −3.453 m/s²
• Normal component: a_n = √(a² − a_t²) = √(29 − 11.92) = 4.133 m/s²
• Radius of curvature: ρ = v²/a_n = 48.35²/4.133 = 565.6 m ≈ 566 m

Position at t = 3 s

• x = u_x t + ½a_x t² = 51.96(3) − ½(2)(9) = 155.88 − 9 = 146.88 m
• y = u_y t + ½a_y t² = 30(3) − ½(5)(9) = 90 − 22.5 = 67.50 m

Answer: ρ ≈ 566 m; position (146.9 m, 67.5 m) from the origin.