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Supporting an injured arm: I. A 650 N person must have her injured arm supported, with the upper arm horizontal and the fore- arm vertical. (See Figure $10.76 .$ ) According to biomedical tables and directmeasurements, her upper arm is 26 $\mathrm{cm}$ long (measured from the shoulder joint), accounts for 3.50$\%$ of her body weight, and has a center of mass 13.0 $\mathrm{cm}$ from her shoulder joint. Her forearm (including the hand) is 34.0 $\mathrm{cm}$ long, makes up 3.25$\%$ of her body weight, and has a center of mass 43.0 $\mathrm{cm}$ from her shoulder joint. (a) Where is the center of mass of the person's arm when it is supported as shown? (b) What weight $W$ is needed to support her arm? (c) Find the horizontal and vertical components of the force that the shoulder joint exerts on her arm.

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Physics 101 Mechanics

Chapter 10

Dynamics of Rotational Motion

Newton's Laws of Motion

Rotation of Rigid Bodies

Equilibrium and Elasticity

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So it's trauma free body diagram of what's going on. We have the Y axis here and the x axis way. We have a pivot point right here, and we have the weight of the upper arm. And then, um, going down here we have the wait of the forearm pointed up is simply the weight we're going to choose counterclockwise to be positive. And we know that this distance here is 17 centimeters. We know that there is a force up here. Um, this is be the force by the shoulder. Ah, this is the weight of the forum weight of the upper arm. And then lastly, we have these distances, and this would be 13 centimeters, and this would also be 13 centimeters. Come on. Yeah. So this would be the complete free body diagram. Now, the question is asking us to find the x coordinates of the X and y co ordinates of the center of mass. So we can say that for part a, the X coordinate for the center of mass will simply be equal to the weight of the upper arm Times 13 centimeters plus the weight of the forearm times 26 centimeters divided by the weight of the upper arm, plus the weight of the forearm. This is going to be equal to 22.8 Newtons, times 13 plus 21.1 Newtons, Times 26 and then this would be divided, of course, by 22.8 Newton's plus 21.1 Nunes. We have that the ex corde it of the center of mass will be equal to 19.2 centimeters to find the Y coordinate of the center of mass. This's going to be equal to the weight of the forearm times 17 centimeters, divided by the weight of the upper arm, plus the weight of the forearm. And so the why cordoned of the center of Mass will be equal to 21.1 Newton's time. 17 centimeters divided by 21.1 Newton's plus 22.8 noons, and we find that the Y coordinate of the center of mass will be equal to 8.2 centimeters. So we can say that ah x, why of the center of mass will be equal to ah, 19.2 comma, 8.2 centimeters come for part B they want us to find the magnitude of Thie. You wait so we can say that Sigma Tau or the sum the net torque is going to be equal to zero. This is not rotating. This will be equal to the weight times 26.0 centimeters. So we're taking existences X distance away from the pivot. This will be minus omega, rather the weight of the forearm times 26.0 centimeters and then minus the weight of the upper arm times 13.0 centimeters. And we find that the wait is going to be equal to the weight of the fort arm, plus 1/2 the weight of the upper arm. So this will be 21.1 Nunes plus 11.4 Newtons, and we find that the wait is going to be 32.5 noons. This will be our magnitude, but it's the end of the solution. Rather, Part C is actually asking us for the force. So we can we know that the force has known ah, horizontal component. And so we know that the magnitude of the force and the Y direction will equal simply the magnitude of the force. So we can say that Sigma F in the white direction equals zero which will equal that force plus the weights plus minus the weight of the upper arm. Ah, minus the weight of the forearm. So we can say that the force will be equal to the weight of the upper arm, plus the weight of the four armed minus the weight. So this will be 22 point eight noons minus 21 rather plus 21.1 Nunes minus 32.5 noons. And we find that the force of the shoulder is going to be equal to 11 point for Newton's. This will be your answer for apart. See, that is the end of the solution. Thank you for watching.

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