Question

You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles. Your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand 64.0cm from your shoulder joint. The deltoid muscle is attached to the humerus 15.0cm from the shoulder joint and makes a 12.0¬[infinity] angle with that bone. If you have set the tension in the cable of the machine to 36.0N on each arm, what is the tension in each deltoid muscle if you simply hold your outstretched arms in place?

Answer 1

To calculate the tension in each deltoid muscle when holding your outstretched arms in place, we can analyze the forces acting on the shoulder joint. By calculating the torque exerted by the deltoid muscle and using the formula for tension, we find that the tension in each deltoid muscle is approximately 0.292 N.

Step-by-step explanation:

To determine the tension in each deltoid muscle when holding your outstretched arms in place, we can analyze the forces acting on the shoulder joint.

First, we need to calculate the torque exerted by the deltoid muscle. Torque is equal to the force applied multiplied by the lever arm.

Given that the tension in the cable of the machine is 36.0N on each arm, the lever arm is 64.0cm from the shoulder joint, and the angle between the deltoid muscle and the humerus is 12.0°, we can calculate the torque using the formula:

Torque = Force imes Lever Arm imes sin(angle)

Substituting the values, we get:

Torque = 36.0N imes 0.64m imes sin(12.0°)

Torque = 0.231 Nm

Since the deltoid muscle is attached to the humerus 15.0cm from the shoulder joint, we can calculate the tension in each deltoid muscle using the formula:

Tension = Torque / (Lever Arm + distance from shoulder joint to attachment point)

Tension = 0.231 Nm / (0.64m + 0.15m)

Tension = 0.231 Nm / 0.79m

Tension = 0.292 N

Therefore, the tension in each deltoid muscle when holding your outstretched arms in place is approximately 0.292 N.

Answer 2

The tension in each deltoid muscle when you hold your outstretched arms in place is approximately 153.6 N.

Convert the angle to radians:

First, we need to convert the deltoid muscle angle from degrees to radians for mathematical calculations. We use the formula:

radians = degrees * pi / 180

In this case, the conversion is:

deltoid_angle_rad = 12.0 * pi / 180 ≈ 0.21 radians

Calculate the moment arm:

The moment arm is the perpendicular distance from the line of action of the force (cable tension) to the axis of rotation (shoulder joint). In this case, it's the distance between the point where the cable attaches to your hand and the point where the deltoid muscle attaches to the humerus, multiplied by the cosine of the deltoid muscle angle:

moment_arm = cable_distance * cos(deltoid_angle_rad)

Plugging in the values, we get:

moment_arm = 64.0 cm * cos(0.21 radians) ≈ 62.5 cm

Calculate the deltoid muscle tension:

The deltoid muscle tension is equal to the cable tension divided by the ratio of the cable distance to the moment arm:

deltoid_tension = cable_tension * (cable_distance / moment_arm)

Substituting the values, we obtain:

deltoid_tension = 36.0 N * (64.0 cm / 62.5 cm) ≈ 153.6 N

Therefore, each deltoid muscle experiences a tension of approximately 153.6 N when you hold your arms outstretched while using the Nautilus machine.