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 76 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 downward force exerted by the lower leg bone on the foot (FB), we need to analyze the forces acting on the system.
Let's consider the following quantities:
- Mass of the person (m): 76 kg
- Length of D: d
- Length of D (twice as long as d): 2d
To estimate FT, we can start by thinking about the equilibrium of forces acting on the person while elevated on the ball of one foot. The main forces at play here are the force of gravity (mg) and the tension in the Achilles tendon (FT).
1. Determine the force of gravity (mg):
The force of gravity acting on the person can be calculated using the formula: mg, where m is the mass of the person and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For this person with a mass of 76 kg:
mg = (76 kg) * (9.8 m/s^2) = 744.8 N
2. Analyze the equilibrium of forces:
Considering the equilibrium of forces in the vertical direction, we know that the tension in the Achilles tendon (FT) must balance the force of gravity (mg).
FT = mg = 744.8 N
Therefore, the estimated tension in the Achilles tendon (FT) is approximately 744.8 N.
Now, let's estimate the downward force exerted by the lower leg bone on the foot (FB).
3. Consider the lever arm effect:
Given the lengths D and d, we can determine the lever arm effect on the downward force exerted by the lower leg bone on the foot (FB).
The lever arm effect states that the longer the lever arm, the less force is required to produce the same torque.
In this case, since D is twice as long as d, the force FB exerted by the lower leg bone on the foot would also be divided into two components, with one component acting at D and the other at d.
Let's assume the forces exerted at D and d are FB_D and FB_d, respectively.
4. Use the lever arm principle:
According to the lever arm principle, the product of the force and its lever arm at two points on a lever is the same.
Therefore, we have: FB_d * d = FB_D * D
Since D is twice as long as d, D = 2d. Substituting this into the equation, we get:
FB_d * d = FB_D * 2d
Simplifying the equation, we find:
FB_d = 2 * FB_D
5. Determine FB_D using torque balance:
To estimate FB_D, we can use the principle of torque balance around the pivot point (the ball of the foot).
Considering the applied force FT at D and the downward force FB_D at D, the torque balance is given by:
FB_D * d = FT * D
Since we know that FT = 744.8 N and D = 2d, we can substitute these values into the equation:
FB_D * d = (744.8 N) * (2d)
Dividing both sides by d, we find:
FB_D = (744.8 N) * 2 = 1489.6 N
6. Calculate FB_d:
Using the relationship FB_d = 2 * FB_D, we can determine FB_d by substituting the value of FB_D:
FB_d = 2 * (1489.6 N) = 2979.2 N
Therefore, the estimated downward force exerted by the lower leg bone on the foot (FB) is approximately 2979.2 N.
To summarize:
- Tension in the Achilles tendon (FT) ≈ 744.8 N
- Downward force exerted by the lower leg bone on the foot (FB) ≈ 2979.2 N