When you stand on your tiptoes, your feet pivot about your ankle. As shown in the figure, the forces on your foot are an upward force on your toes from the floor, a downward force on your ankle from the lower leg bone, and an upward force on the heel of your foot from your Achilles tendon. Suppose a 61 woman stands on tiptoes with the sole of her foot making a 25 angle with the floor. Assume that each foot supports half her weight.
Sorry I can't add a picture to this b/c cramster doesn't have my book but....
From the toe to the ankle pivot is 15cm and from the toe to the Achillies tendon it is 20cm.
a)What upward force does the Achilles tendon exert on the heel of her foot N?
b)The tension in the Achilles tendon will cause it to stretch. If the Achilles tendon is 15 long and has a cross-section area of 110 , by how much will it stretch under this force?
927
Many of your numbers require dimensions. For example, what do you mean by "a 61 woman"?
883N
To solve this problem, we need to analyze the forces acting on the foot and apply equilibrium conditions.
a) The upward force exerted by the Achilles tendon on the heel of the foot can be found by summing the vertical forces acting on the foot. Let's denote this force as F_Achilles. Since each foot supports half of the woman's weight, the weight acting on each foot is (1/2)*(61 kg)*g, where g is the acceleration due to gravity.
The weight of the woman acting on one foot can be split into two components: one acting vertically down from the ankle pivot and another acting vertically down from the Achilles tendon. Using trigonometry, we can express these components as follows:
Vertical component from the ankle pivot = (1/2)*(61 kg)*g * cos(25°)
Vertical component from the Achilles tendon = (1/2)*(61 kg)*g * sin(25°)
Since the foot is in equilibrium, the vertical forces must balance each other. Therefore, the upward force exerted by the Achilles tendon can be found by equating the two components:
F_Achilles = (1/2)*(61 kg)*g * sin(25°) / cos(25°)
b) To determine the stretch in the Achilles tendon, we need to apply Hooke's law, which states that the force exerted on a spring-like material is proportional to the amount it is stretched or compressed. Hooke's law can be written as F = k * ΔL, where F is the force, k is the spring constant, and ΔL is the change in length.
In this case, the force exerted by the Achilles tendon is F_Achilles, and the original length of the tendon is 15 cm. To find the change in length, we rearrange Hooke's law equation:
ΔL = F_Achilles / k
To find the spring constant k, we need to know the Young's modulus of elasticity for the Achilles tendon material. Without this information, it is not possible to calculate the stretch in the tendon.
Therefore, to fully solve part b of the problem, we would need the Young's modulus of elasticity of the Achilles tendon material.