Question

11.24 ·· bio a good workout. you are doing exercises on a nautilus machine in a gym to strengthen your deltoid (shoulder) mus- cles. your arms are raised vertically and can pivot around the shoulder joint, and you grasp the cable of the machine in your hand 64.0 cm from your shoulder joint. the deltoid muscle is attached to the humerus 15.0 cm from the shoulder joint and makes a 12.0° angle with that bone (fig. e11.24). if you have set the tension in the cable of the machine to 36.0 n on each arm, what is the tension in each deltoid muscle if you simply hold your outstretched arms in place? (hint: start by making a clear free - body diagram of your arm.) figure e11.24

Explanation:

Step1: Calculate the angle for the tension force

The angle between the tension force \(T\) (from the cable) and the lever arm (distance from shoulder joint to hand) is \(\theta_T = 180^{\circ}- 125^{\circ}=55^{\circ}\)

Step2: Set up the torque equilibrium equation

The torque due to the cable tension \(\tau_T\) and the torque due to the deltoid muscle tension \(\tau_m\) must balance. The torque formula is \(\tau = rF\sin\theta\). Let \(r_T=64.0\space cm = 0.64\space m\), \(F_T = 36.0\space N\), \(r_m=15.0\space cm=0.15\space m\), and \(\theta_m = 12.0^{\circ}\). At equilibrium \(\tau_T=\tau_m\), so \(r_TF_T\sin\theta_T=r_mF_m\sin\theta_m\)

Step3: Solve for the deltoid muscle tension \(F_m\)

$$
LATEXBLOCK0
$$

We know that \(\sin(55^{\circ})\approx0.819\) and \(\sin(12^{\circ})\approx0.208\)

$$
LATEXBLOCK1
$$

Answer:

The tension in each deltoid muscle is approximately \(606\space N\)