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ps12sol

# ps12sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Problem Set 12 Static Equilibrium and Three Dimensional Rotations, Solutions Problem 1: Static Equilibrium The Ankle A person of mass m = 75 kg is crouching with his/her weight evenly distributed on both tiptoes. The forces on the skeletal part of the foot are shown in the diagram. In this position, the tibia acts on the foot with a force F ! of magnitude F = F ! and which makes an unknown angle ! with the vertical. This force acts on the ankle a horizontal distance s = 4.8 cm from the point where the foot contacts the floor. The Achilles tendon is under considerable tension T ! and makes a given angle ! = 37 0 with the horizontal. The tendon acts on the ankle a horizontal distance b = 6.0 cm from the point where the tibia acts on the foot. You may ignore the weight of the foot. Let g = 9.8 m ! s -2 be the gravitational constant. In this problem you will express your answers symbolically. You may want to substitute in numbers if you have the time. a) Find the magnitude of the tension in the Achilles tendon, T ! T ! . b) Find the magnitude, F ! F ! , and the angle, ! , of the tibia force on the ankle. Solution: From the description of the problem, you are trying to determine a symbolic expression for the forces that are distributed over a foot that is in static equilibrium, and hence the two laws of static equilibrium apply:

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(1) The sum of the forces acting on the rigid body is zero, total 1 2 = + + !!! = F F F 0 ! ! ! ! (1) (2) The vector sum of the torques about any point S in a rigid body is zero, total ,1 ,2 S S S ! ! ! = + + """ = 0 ! ! ! ! (2) In order to apply these laws you must do the following: i) Determine precisely which part of the body you will choose to consider as the system on which the forces are acting, ii) Because the foot is not an isolated body, you must consider exactly which forces are acting on the foot, and at which points they act. iii) Determine which point about which to calculate torques, and a choice for positive direction for the torque (counterclockwise in the above diagram). Remember that if you choose a point where a force acts then that force has zero torque about that point. You should see if you can find some special point which would simplify your torque calculation. iv) When calculating torque about a chosen point, you also need to decide for each force whether the given information of the problem makes it easier to compute the moment arm of the force about your chosen point or the perpendicular component of the force with respect to a line drawn from your chosen point to the point where the force acts. Design a Strategy: System: Choose the foot as the system noting that the tibia bone and the Achilles tendon are not part of the system. Forces: There are three forces acting on the foot. The normal force of the floor acts on the foot. You can actually determine the normal force by considering the entire body. Since the weight is evenly distributed on the two feet, the normal force on one foot is equal to half the weight, or N = 1 2 ( ) mg . The Achilles tendon exerts a force of unknown magnitude T ! T !
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ps12sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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