13.59. Model: r Model the beam as a rigid body. For the beam not to fall over, it must be both in translational equir librium ( Fnet = 0 N ) and rotational equilibrium (τ net = 0 N m ). Visualize:
The boy walks along the beam a distance x, measured from the left end of the beam. There are four forces acting on r r the beam. F1 and F2 are from the two supports, wb is the weight of the beam, and wB is the weight of the boy. Solve: We pick our pivot point on the left end through the first support. The equation for rotational equilibrium is –wb (2.5 m) + F2 (3.0 m) – wBx = 0 N m –(40 kg)(9.80 m/s2)(2.5 m) + F2 (3.0 m) – (20 kg)(9.80 m/s2)x = 0 N m The equation for translation equilibrium is
∑F
y
= 0 N = F1 + F2 − wb − wB
⇒ F1 + F2 = wb + wB = ( 40 kg + 20 kg)(9.8 m /s 2 ) = 588 N Just when the boy is at the point where the beam tips, F1 = 0 N. Thus F2 = 588 N. With this value of F2, we can simplify the torque equation to: –(40 kg)(9.80 m/s2)(2.5 m) + (588 N)(3.0 m) – (20 kg)(9.80 m/s2)x = 0 N m
⇒ x = 4.0 m Thus, the distance from the right end is 5.0 m – 4.0 m = 1.0 m.