Suppose the hand in holds a 20 kg mass . What force, FM, is required of the deltoid muscle? Assume the mass is 50 cm from the shoulder joint, the force of tension in the deltoid is directed 15° to the horizontal, and the mass of the arm alone is 3.8 kg.

To find the force required of the deltoid muscle (FM), we can use the principle of torque and equilibrium.

1. First, let's calculate the torque caused by the weight of the mass and the arm. Torque is given by the formula:

Torque = Force × distance × sin(θ)

Where:
- Force is the force of tension in the deltoid (FM) that we want to find.
- Distance is the distance from the shoulder joint to the mass, which is given as 50 cm (or 0.5 meters).
- θ is the angle between the force of tension and the horizontal, given as 15°.
- sin(θ) is the sine of the angle in radians.

2. We also need to consider the torque caused by the weight of the arm alone. The torque due to the arm is given by the formula:

Torque_arm = (mass of the arm) × (gravitational acceleration) × (distance from the shoulder joint to the center of mass of the arm)

Where:
- The mass of the arm is given as 3.8 kg.
- The gravitational acceleration is approximately 9.8 m/s².
- The distance from the shoulder joint to the center of mass of the arm is not provided, but we can assume it is at the midpoint of the arm, which is half of the distance from the shoulder joint to the mass (25 cm or 0.25 meters).

3. To maintain equilibrium, the clockwise torque caused by the weight of the mass and arm should be balanced by the counterclockwise torque caused by the deltoid muscle.

Clockwise torque = Counterclockwise torque

Torque_mass + Torque_arm = Torque_deltoid

4. Substitute the values into the equations and solve for FM.

Torque_mass = (mass of the mass) × (gravitational acceleration) × (distance from the shoulder joint to the mass) × sin(θ)

Torque_deltoid = FM × (distance from the shoulder joint to the mass) × sin(θ)

FM = (Torque_mass + Torque_arm) / (distance from the shoulder joint to the mass) × sin(θ)

FM = [(mass of the mass) × (gravitational acceleration) × (distance from the shoulder joint to the mass) × sin(θ) + (mass of the arm) × (gravitational acceleration) × (distance from the shoulder joint to the center of mass of the arm)] / (distance from the shoulder joint to the mass) × sin(θ)

Now, plug in the given values and solve the equation.