The Achilles tendon is attached to the rear of the foot. Assume that a person has a mass of 53 kg and that D is twice as long as d. When a person elevates himself just barely off the floor on the "ball of one foot," estimate the tension in the Achilles tendon (pulling upward).
The Achilles tendon is attached to the rear of the foot. A person elevates himself just barely off the floor on the "ball of one foot." Assume the person has a mass of 60 kg and D is twice as long as d. Find the tension FT in the Achilles tendon (pulling upward).
To estimate the tension in the Achilles tendon when a person elevates themselves just barely off the floor on the ball of one foot, we can use the concept of torque. Torque is the product of force and the lever arm distance.
Let's break down the problem step-by-step:
Step 1: Determine the force exerted by the person's weight
The force exerted by the person's weight can be calculated using the formula F = mg, where m is the mass and g is the acceleration due to gravity. Given that the person's mass is 53 kg, and accelaration due to gravity is approximately 9.8 m/s^2, we can calculate the force exerted.
F = (53 kg) * (9.8 m/s^2) = 519.4 N
Step 2: Calculate the lever arm distances, d and D
According to the problem statement, we are given the length of the shorter lever arm as d. Let's say we take d as the distance between the center of mass of the body and the ball of the foot. Given that D is twice as long as d, we can write the equation: D = 2d.
Step 3: Calculate the torque
Torque is calculated by multiplying the force by the lever arm distance. Since there are two lever arm distances involved (d and D), we will calculate the torque for both and sum them up.
T_d = F * d
T_D = F * D
Step 4: Calculate the tension in the Achilles tendon
The tension in the Achilles tendon can be considered to act at the ball of the foot, pulling upwards. Since there are two torques acting in opposite directions, the tension in the Achilles tendon can balance out the torque due to the person's weight.
Tension = T_D - T_d
Step 5: Substitute the values to calculate the tension
Substituting the values of F, d, and D, we can calculate the tension in the Achilles tendon.
Tension = (F * D) - (F * d)
= (519.4 N) * (2d) - (519.4 N) * d
= 519.4 N * d
So, the tension in the Achilles tendon, pulling upward, is estimated to be 519.4 N times the length of the shorter lever arm (d).
To estimate the tension in the Achilles tendon, we need to consider the equilibrium of forces acting on the person's body while they elevate themselves on the "ball of one foot".
Let's analyze the forces involved:
1. Weight of the person: The person's weight acts downward and can be determined using the mass and acceleration due to gravity. Assuming g as approximately 9.8 m/s², the weight can be calculated as weight = mass × g.
2. Tension in the Achilles tendon: The Achilles tendon provides an upward force to counteract the downward force of the person's weight. This force can be considered as the tension in the Achilles tendon.
3. Force from the ground: When a person elevates themselves on the ball of one foot, the ground exerts a normal force upward to support the person's weight.
Now, let's break down the solution step-by-step:
Step 1: Calculate the weight of the person
Given the mass of the person is 53 kg, we can calculate the weight as follows:
Weight = mass × g = 53 kg × 9.8 m/s² = 519.4 N (rounded to one decimal place)
Step 2: Analyze the forces acting on the person
Since the person is barely off the floor, we assume their body is in equilibrium. This implies that the upward forces and downward forces cancel each other out. Therefore,
Force from the Achilles tendon = Weight + Force from the ground
Step 3: Determine the force from the ground
To estimate the force from the ground, we need to consider the lever system formed by the foot. Given that D is twice as long as d, we can use the concept of torque.
The torque about the ankle joint is given by: T = force × perpendicular distance
Since the foot is elevated on the ball and the heel remains on the ground, the perpendicular distances are d/2 and D/2, respectively, where d and D represent the distances from the ankle joint to the ball of the foot and the heel.
For equilibrium, the torques on both sides should cancel each other:
Torque from the Achilles tendon = Torque from the ground
Step 4: Estimate the tension in the Achilles tendon
The torque from the Achilles tendon is given by: Tension × (D/2)
By equating the torque from the Achilles tendon to the torque from the ground, we can solve for the tension.
Once you have the equation, you can use algebraic manipulation to rearrange the terms and find the estimated tension in the Achilles tendon.
Please provide the values of d and D, and I can help you calculate the tension in the Achilles tendon.