The arm in the figure below weighs 44.1 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force Ft in the deltoid muscle and the force Fs exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown. (Enter your answers to at least the nearest newton.)Ft = Enter a number. NFs = Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. NAn arm is extending horizontally to the right with a cutout that shows muscles in the upper arm. A horizontal dashed line extends along the bottom of the muscles and another along the top. Point O is labeled at the left end of the arm on the lower horizontal line. An unlabeled point lies 0.080 m horizontally to the right of point O. Point A lies 0.290 m horizontally to the right of point O. Three forces are acting on the arm.A force vector Fs acts down and right on point O at an acute angle 𝜃 below the horizontal.A force vector Ft acts up and left on the unlabeled point at an angle 12° above the horizontal. Vector Ft is longer than vector Fs.A force vector Fg acts vertically down on point A. Vector Fg is shorter than vector Fs
Question
The arm in the figure below weighs 44.1 N. The force of gravity acting on the arm acts through point A. Determine the magnitudes of the tension force Ft in the deltoid muscle and the force Fs exerted by the shoulder on the humerus (upper-arm bone) to hold the arm in the position shown. (Enter your answers to at least the nearest newton.)Ft = Enter a number. NFs = Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. NAn arm is extending horizontally to the right with a cutout that shows muscles in the upper arm. A horizontal dashed line extends along the bottom of the muscles and another along the top. Point O is labeled at the left end of the arm on the lower horizontal line. An unlabeled point lies 0.080 m horizontally to the right of point O. Point A lies 0.290 m horizontally to the right of point O. Three forces are acting on the arm.A force vector Fs acts down and right on point O at an acute angle 𝜃 below the horizontal.A force vector Ft acts up and left on the unlabeled point at an angle 12° above the horizontal. Vector Ft is longer than vector Fs.A force vector Fg acts vertically down on point A. Vector Fg is shorter than vector Fs
Solution
To solve this problem, we need to apply the principles of equilibrium. For an object to be in equilibrium, the sum of the forces and the sum of the torques acting on it must be zero.
Given: Weight of the arm (Fg) = 44.1 N Distance from point O to the point where Ft acts = 0.080 m Distance from point O to point A = 0.290 m Angle of Ft = 12 degrees
- Calculate the torque due to the weight of the arm:
Torque is given by the equation Torque = Force x Distance x sin(angle). The force of gravity acts vertically downward, so the angle between the force of gravity and the line from point O to point A is 90 degrees. Therefore, sin(90) = 1.
Torque due to Fg = Fg x Distance = 44.1 N x 0.290 m = 12.789 N.m
- Calculate the tension force Ft:
The arm is in equilibrium, so the torque due to Ft must balance the torque due to Fg. Therefore, Ft x 0.080 m x sin(180 - 12) = 12.789 N.m.
Solving for Ft gives Ft = 12.789 N.m / (0.080 m x sin(168)) = 162.5 N
- Calculate the shoulder force Fs:
The arm is in equilibrium, so the sum of the forces in the vertical direction must be zero. The forces in the vertical direction are -Fg, Ftsin(12), and Fssin(θ). Therefore, Fssin(θ) = Fg - Ftsin(12).
We don't know the angle θ, but we know that the sum of the forces in the horizontal direction must also be zero. The forces in the horizontal direction are Fscos(θ) and Ftcos(12). Therefore, Fscos(θ) = Ftcos(12).
Dividing these two equations gives tan(θ) = (Fg - Ftsin(12)) / (Ftcos(12)). Solving for θ gives θ = atan((Fg - Ftsin(12)) / (Ftcos(12))) = 20.5 degrees.
Substituting θ back into the equation for Fs gives Fs = (Fg - Ft*sin(12)) / sin(20.5) = 213.7 N.
So, the tension force Ft in the deltoid muscle is 162.5 N and the force Fs exerted by the shoulder on the humerus is 213.7 N.
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