When a person stands on tiptoe (a strenuous position), the position of the foot is as shown in Figure P8.16a. The total gravitational force on the body, Fg, is supported by the force n exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P8.16b, where T is the force exerted by the achilles tendon on the foot and R is the force exerted by the tibia on the foot. Find the values of T, R, and θ when Fg = 765 N.
To find the values of T, R, and θ when Fg = 765 N, we can analyze the forces acting on the foot and solve for each unknown force.
1. Start by drawing a free-body diagram of the forces acting on the foot. We have the following forces:
- Fg: gravitational force acting vertically downward.
- n: force exerted by the floor on the toes, perpendicular to the floor.
- T: force exerted by the Achilles tendon on the foot, at an angle θ.
- R: force exerted by the tibia on the foot, perpendicular to the tibia.
2. Since the foot is in static equilibrium, the sum of the forces in both the x and y directions must be zero. Let's consider the forces in the y direction:
ΣFy = n - Fg = 0
From this equation, we can solve for n:
n = Fg
Substituting Fg = 765 N, we have:
n = 765 N
3. Now let's analyze the forces in the x direction:
ΣFx = T - R = 0
From this equation, we can solve for R:
R = T
4. To find the value of T, we need to analyze the torques acting on the foot. Torque is equal to the force applied multiplied by the lever arm.
Since the foot is rotating about the Achilles tendon, the torque equation becomes:
Στ = 0
Taking the torque about the Achilles tendon as the pivot point:
τT + τR - τn = 0
Here, τT = T * d, τR = R * d, and τn = n * l, where d is the distance between the Achilles tendon and the pivot point, and l is the distance between the toes and the pivot point.
Since the foot is at the tip-toe position, the forces R and T are both perpendicular to the lever arm d.
Therefore, we have:
T * d + R * d - n * l = 0
Substituting R = T and n = Fg, and rearranging the equation, we get:
T * d + T * d - Fg * l = 0
Simplifying further:
2 * T * d = Fg * l
Now, we can solve for T:
T = (Fg * l) / (2 * d)
Substituting Fg = 765 N, l = length of the foot, and d = distance between the Achilles tendon and the pivot point, we can calculate the value of T.
5. Finally, to find the value of θ, we can use trigonometry. In the mechanical model shown in Figure P8.16b, sin(θ) = R / T.
Since R = T, we have:
sin(θ) = 1
Taking the inverse sine of both sides, we get:
θ = sin^(-1)(1)
θ = 90 degrees
Therefore, when Fg = 765 N, the values are:
T = (Fg * l) / (2 * d)
R = T
θ = 90 degrees
To find the values of T, R, and θ when Fg = 765 N, we can use the principles of static equilibrium. In this case, the sum of the forces in the vertical direction must equal zero, and the sum of the torques around any point must also equal zero.
Step 1: Resolve the forces in the vertical direction:
Since the person is standing on tiptoe, we can assume that the force n exerted by the floor on the toes of one foot is acting vertically upward.
The weight or gravitational force, Fg, is acting vertically downward.
Summing up the forces in the vertical direction:
n - Fg = 0
n = Fg
Step 2: Analyze the torques around the pivot point (achilles tendon):
In the mechanical model shown in Figure P8.16b, the pivot point is the achilles tendon. The forces acting on the foot are the force T exerted by the achilles tendon on the foot and the force R exerted by the tibia on the foot.
The torque exerted by the force T is T * (d/2) * sinθ, where d is the distance between the point of force application and the pivot point (achilles tendon).
The torque exerted by the force R is R * (d) * sinθ.
Since the torques around the pivot point must balance, we have:
T * (d/2) * sinθ + R * (d) * sinθ = Fg * h,
where h is the distance between the pivot point (achilles tendon) and the point where the force n is applied.
Step 3: Solve for T, R, and θ:
To find the values of T, R, and θ, we need additional information such as the distances d and h. Without this information or further assumptions or data, it is not possible to determine the values of T, R, and θ.
Please provide the values of d and h if available or any other relevant information, so we can proceed with the calculation.