Question

Interactive Exercises 9.04: Center of Mass (Robotic Arm) Robotic arms have been used on factory assembly lines for many years, but new research into the direct control of robotic arms by humans is creating an opportunity for disabled individuals to regain some degree of lost mobility-see Fig. 9.4.1. In this application, you are to analyse a simple model of the upper part of a robotic arm to find the location of its center of mass. The model is shown in Fig. 9.4.2. Let's model the shoulder as a uniform cube of mass M = 110 kg and edge length = 0.25 m. Connected to the shoulder is the upper arm, which is a uniform rod (cylinder) of mass and length L. Connected to the opposite end of the upper arm is the elbow joint, which is a uniform spherical shell of outer diameter 1/2. If it were solid, the sphere would have a mass 2M, but the sphere has had an inner spherical volume of radius 1/8 hollowed out. The same model is depicted in the simulation (linked below). In the simulation, the value of the ratio rlL is adjustable. The simulation also indudes an x axis that is directed along the length of the arm. Interactive Figure 9.4.1: A scientist at the Institute for Automation in Bremen, Germany demonstrates an ability to use a robotic arm to lift a book from a shelf and place it on a table. The man uses a joystick just under his chin to control the arm. (Ingo Wagner/dpa/picture-alliance/News.com) Shoulder Elbow Upper arm 1 2

Answer 1

1) Outer radius L/2 0.25 2 2 Ri -0.125m Inner radius - ulos L/2 4 R₂ 0.25 4 = 0.0625 m Density = 2 M 4 TT R 3 = 3 2 X 11 4 x3.14 x (0.125) 3 = 2690.4 Kg/m?

3 2) Volume 2 Sheld = 1 (R,'-R R?) M Shell Ру 3 = 2690.4 X 3-14 (10-125) -(0-0625 = 19.25 kg 3) X COM 2M16) + M(L) +Mthe (72/4) 2 M4 M + Mshell = M( 42 ) + 2M ( + 2) + Mistell (2 x 1 ) 3M t M shell 1110.25)+ 22 (0,25) +19:25(x0-25 33 + 19.25 2.75 +5.5 +16.84375 = 52.25 0.48 m