Calculate the magnitude of the force FM required of the "deltoid" muscle to hold up the outstretched arm shown in the figure below. The total mass of the arm is 2.8 kg and θ = 11°.
d=12cm D=24cm
To calculate the magnitude of the force FM required to hold up the outstretched arm, we can use the principles of torque and equilibrium.
First, let's understand the situation and the variables involved:
- The arm is outstretched, and we are interested in the force required to hold it in that position.
- The total mass of the arm is given as 2.8 kg.
- d = 12 cm represents the distance from the pivot point to the center of the deltoid muscle.
- D = 24 cm represents the distance from the pivot point to the weight of the arm.
Now, let's approach the problem step by step:
1. Draw a free-body diagram of the arm, showing all the forces acting on it.
- There are two main forces: the force FM generated by the deltoid muscle and the force of gravity acting on the mass of the arm.
2. Identify the pivot point or the point where the arm rotates.
- In this case, the pivot point is not explicitly given, but we can assume it to be the point of rotation of the shoulder joint where the deltoid muscle is attached.
3. Apply the principle of equilibrium.
- For the arm to be held up in a static position without any rotational acceleration, the sum of the torques about any point must be zero.
4. Calculate the torque due to the force of gravity.
- The torque due to the force of gravity can be calculated as the product of the gravitational force and the perpendicular distance from the point of rotation to the line of action of the force.
- The torque due to the force of gravity (τg) can be given by τg = mgDsinθ, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), D is the distance from the pivot point to the weight, and θ is the angle mentioned in the problem.
5. Calculate the torque due to the force FM.
- The torque due to the force FM can be calculated as the product of the force and the perpendicular distance from the point of rotation to the line of action of the force.
- The torque due to the force FM (τm) can be given by τm = FMd.
6. Set up the equilibrium equation.
- Since the torques must balance each other for equilibrium, we can set up an equation as follows: τm = τg.
7. Substitute the values into the equation and solve for FM.
- Plug in the known values: τm = FMd and τg = mgDsinθ.
- Equate the two expressions and solve for FM: FMd = mgDsinθ.
- Divide both sides by d to isolate FM: FM = mgDsinθ / d.
8. Calculate FM.
- Now, substitute the given values of mass (m = 2.8 kg), angle (θ = 11°), distances (d = 12 cm and D = 24 cm), and the acceleration due to gravity (g = 9.8 m/s^2) into the equation obtained in step 7.
- Convert all the distances to meters: d = 0.12 m and D = 0.24 m.
- Calculate FM: FM = (2.8 kg)(9.8 m/s^2)(0.24 m)sin(11°) / 0.12 m.
By following these steps, you can calculate the magnitude of the force FM required of the deltoid muscle to hold up the outstretched arm.