The Achilles tendon is attached to the rear of the foot as shown in the figure.(Figure 1) When a person elevates himself just barely off the floor on the "ball of one foot," estimate the tension FT in the Achilles tendon (pulling upward). Assume the person has a mass of 78 kg and D is twice as long as d. and Estimate the (downward) force FB exerted by the lower leg bone on the foot.
To estimate the tension in the Achilles tendon (FT) and the force exerted by the lower leg bone on the foot (FB), we can use some basic principles of mechanics and apply them to the given situation.
Let's break down the problem into smaller steps:
1. Identify the forces involved:
- Tension force in the Achilles tendon (FT): This is the force pulling upward on the foot.
- Force exerted by the lower leg bone on the foot (FB): This is the force pushing downward on the foot.
- Weight of the person (mg): The force due to gravity acting on the person's mass. In this case, it's the force pulling downward on the person.
2. Establish the equilibrium condition:
When a person elevates themselves just barely off the floor on the "ball of one foot," we can assume that the foot is in equilibrium. This means that the sum of all forces acting on the foot is zero:
ΣF = 0
3. Resolve forces along the vertical axis:
Since the person is not accelerating vertically, the sum of forces along the vertical axis should be zero. From this, we can write:
FT - FB - mg = 0
4. Determine the relationship between the distances:
The problem states that "D is twice as long as d." Let's assume that d is the distance from the attachment point of the Achilles tendon to the center of mass of the foot. In that case, D would be twice that distance, or 2d.
Now, we can solve the equations:
From step 3, we have FT - FB - mg = 0.
Rearranging the equation, we get FT = FB + mg.
Next, let's consider the weight of the person:
Weight of the person (mg) = mass (m) * gravitational acceleration (g)
m = 78 kg (as given)
g = 9.8 m/s² (acceleration due to gravity)
By multiplying the mass and gravitational acceleration, we find:
mg = 78 kg * 9.8 m/s² = 764.4 N
Substituting this value in our equation:
FT = FB + 764.4 N
Now, let's consider the relationship between the distances:
D = 2d
Since the Achilles tendon is attached to the rear of the foot, the distance from the attachment point to the center of mass of the foot is half of the total foot length. Therefore:
2d = Foot length
At this point, we would need the specific measurements of the foot length provided in the figure to proceed with the calculation. Unfortunately, Figure 1 is not visible or described in the question text. So without the exact foot length value, we cannot give a numeric estimate for FT or FB.
To complete the calculation, you would need to input the foot length value into the equation 2d = Foot length, and then substitute it to solve for FT and FB using the equation FT = FB + 764.4 N.