The Achilles tendon is attached to the rear of the foot as shown in (Figure 1) . A person elevates himself just barely off the floor on the "ball of one foot." Assume the person has a mass of 65kg and D is twice as long as d.
Find the tension FT in the Achilles tendon (pulling upward).
Find the (downward) force FB exerted by the lower leg bone on the foot.
We can solve this problem using the principles of static equilibrium.
First, let's label the given quantities:
Mass of person (m) = 65 kg
Gravitational acceleration (g) = 9.81 m/s²
Ratio of D to d = 2
When the person is in equilibrium, the upward force from the Achilles tendon (FT) and the downward force from the lower leg bone (FB) should balance the downward gravitational force (Fg) acting on the person. Also, the torques about any point should sum to zero.
1. Balancing vertical forces:
FT + FB = Fg
FT + FB = mg
2. Balancing torques:
For this, we can consider torques about point B, where the FB force acts. The distances of the Achilles tendon force (FT) and gravitational force (mg) are d and D, respectively, from point B.
Torque due to FT = d * FT
Torque due to mg = D * mg
Since D is twice as long as d, D = 2d. Equating the torques in equilibrium, we get:
d * FT = 2d * mg
Dividing both sides by d, we get:
FT = 2mg
Now we can plug in the values to find the tension FT in the Achilles tendon:
FT = 2 * (65 kg) * (9.81 m/s²)
FT ≈ 1273 N
This is the tension in the Achilles tendon, pulling upwards.
Now, we can use the vertical force equation to find FB:
FB = mg - FT
FB = (65 kg * 9.81 m/s²) - 1273 N
FB ≈ 390 N
This is the downward force exerted by the lower leg bone on the foot.
So, the tension in the Achilles tendon is 1273 N (upwards), and the force exerted by the lower leg bone on the foot is 390 N (downwards).
To find the tension FT in the Achilles tendon and the downward force FB exerted by the lower leg bone on the foot, we can use the principles of equilibrium. This means that the net force and net torque acting on the system must be zero.
1. Tension FT in the Achilles tendon (pulling upward):
To find the tension in the Achilles tendon, we need to consider the torque balance around the ankle joint. The torque produced by the weight of the person (W) about the ankle joint is counteracted by the tension in the Achilles tendon.
Let's break down the process step by step:
Step 1: Calculate the weight (W) of the person.
The weight (W) is given by the mass (m) multiplied by the acceleration due to gravity (g). In this case, as the person's mass is 65 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we have:
W = m * g = 65 kg * 9.8 m/s^2 = 637 N (rounded to the nearest whole number).
Step 2: Determine the torque due to the tension in the Achilles tendon.
The torque (τ) is calculated by multiplying the force (FT) in the Achilles tendon by the distance (d) between the Achilles tendon and the ankle joint. Since it is given that D is twice as long as d, we can express the torque as:
τ = FT * d = FT * (1/2)D.
Step 3: Set up the torque balance equation.
In equilibrium, the torque produced by the weight (τw) should be equal and opposite to the torque produced by the tension in the Achilles tendon (τt). Thus,
τw = τt.
Step 4: Solve the torque balance equation for FT.
Substituting the values we obtained:
W * d = FT * (1/2)D.
Plugging in the values of W (637 N), d, and D, you can solve for FT.
2. Downward force FB exerted by the lower leg bone on the foot:
To find the downward force exerted by the lower leg bone on the foot, we need to consider the vertical equilibrium.
Let's break down the process step by step:
Step 1: Set up the vertical equilibrium equation.
In equilibrium, the sum of the vertical forces should be zero. Thus, the upward force FT in the Achilles tendon should be equal and opposite to the downward force FB exerted by the lower leg bone. Mathematically,
FT = FB.
Step 2: Solve for FB.
Since we have already solved for FT in the previous step, the value of FT can be substituted into the equation to find the downward force FB.
By following these steps and plugging in the appropriate values for your specific scenario, you can find the tension FT in the Achilles tendon and the downward force FB exerted by the lower leg bone on the foot.