Answer: In Exercises 6366, the torque about the origin O due to a force F acting on an

Chapter 12, Problem 65

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In Exercises 6366, the torque about the origin O due to a force F acting on an object with position vector r is the vector quantity = r F. If several forces Fj act at positions rj , then the net torque (units: N-m or lb-ft) is the sum = rj Fj Torque measures how much the force causes the object to rotate. By Newtons Laws, is equal to the rate of change of angular momentum. Let be the net torque about O acting on the robotic arm of Figure 23. Assume that the two segments of the arms have mass m1 and m2 (in kilograms) and that a weight of m3 kg is located at the endpoint P. In calculating the torque, we may assume that the entire mass of each arm segment lies at the midpoint of the arm (its center of mass).Show that the position vectors of the masses m1, m2, and m3 are r1 = 1 2 L1(sin 1i + cos 1j) r2 = L1(sin 1i + cos 1j) + 1 2 L2(sin 2i cos 2j) r3 = L1(sin 1i + cos 1j) + L2(sin 2i cos 2j) Then show that = g L1 1 2 m1 + m2 + m3 sin 1 + L2 1 2 m2 + m3 sin 2 k where g = 9.8 m/s2. To simplify the computation, note that all three gravitational forces act in the j direction, so the j-components of the position vectors ri do not contribute to the torque.