The drawing shows an outstretched arm (0.61 m in length) that is parallel to the floor. The arm is pulling downward against the ring attached to the pulley system, in order to hold the 126-N weight stationary. To pull the arm downward, the latissimus dorsi muscle applies the force vector M in the drawing, at a point that is 0.069 m from the shoulder joint and oriented at an angle of 29°. The arm has a weight of 53 N and a center of gravity (cg) that is located d = 0.29 m from the shoulder joint. Find the magnitude of vector M .
52.1
To find the magnitude of vector M, we need to consider the forces acting on the arm.
First, let's calculate the torque caused by the weight of the arm about the shoulder joint. Torque is calculated as the product of the force and the perpendicular distance from the line of action of the force to the point of rotation (or the shoulder joint in this case).
The weight of the arm (53 N) creates a torque about the shoulder joint, given by the equation: Torque = Weight * Distance.
Torque_about_shoulder = Weight_of_arm * Distance_to_cg
Torque_about_shoulder = 53 N * 0.29 m
Next, let's consider the torque required to hold the weight stationary. Since the arm is in equilibrium, the sum of the torques must be zero. Thus, the torque created by the weight of the arm must be balanced by the torque created by the force vector M.
Torque_balance = Torque_about_shoulder - Torque_created_by_M
Since the arm is parallel to the floor, the torque created by the weight of the arm is counterclockwise, while the torque created by M is clockwise. So we need to subtract the torque created by M from the torque about the shoulder joint:
Torque_balance = Torque_about_shoulder - Torque_created_by_M
0 = Torque_about_shoulder - Torque_created_by_M
Since torque is given by the equation: Torque = Force * Distance * sin(θ), we can rewrite the equation replacing torque with the corresponding forces and distances:
0 = Weight_of_arm * Distance_to_cg * sin(θ) - M * Distance_to_shoulder * sin(Ø)
Simplifying the equation:
Weight_of_arm * Distance_to_cg * sin(θ) = M * Distance_to_shoulder * sin(Ø)
Now we can rearrange the equation to solve for M:
M = (Weight_of_arm * Distance_to_cg * sin(θ)) / (Distance_to_shoulder * sin(Ø))
Plugging in the given values:
M = (53 N * 0.29 m * sin(29°)) / (0.069 m * sin(Ø))
Now, we need the value of sin(Ø) to proceed further. The drawing does not provide the angle Ø, so we cannot calculate the exact value of M without that information.