The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (see figure (a)). The forces on the lower leg when the leg is extended are modeled as in figure (b), where vector T is the force of tension in the tendon, vector w is the force of gravity acting on the lower leg, and vector F is the force of gravity acting on the foot. Find (the magnitude of) vector T when the tendon is at an angle θ of 25.0° with the tibia, assuming that w = 31.0 N, F = 11.3 N, and the leg is extended at an angle θ of 40.0° with the vertical. Assume that the center of gravity of the lower leg is at its center and that the tendon attaches to the lower leg at a point one-fifth of the way down the leg.
N
Alas...no figure a
To find the magnitude of vector T, we can decompose the forces acting on the lower leg and solve for the tension in the tendon.
First, let's identify the known values:
- w = 31.0 N (force of gravity acting on the lower leg)
- F = 11.3 N (force of gravity acting on the foot)
- θ = 40.0° (angle of the leg with the vertical)
- The center of gravity is at the center of the lower leg.
- The tendon attaches to the lower leg at a point one-fifth of the way down the leg.
Next, let's draw a diagram to understand the forces involved. We have a right triangle with the leg and the vertical, where the angle is θ = 40.0°.
|
|
|
|\
| \
T | \
| \
| \
| \
| \
|_______\
w
From the diagram, we can see that the force of gravity acting on the lower leg (w) can be decomposed into horizontal and vertical components. The vertical component is w⋅cos(θ) and the horizontal component is w⋅sin(θ).
To find the tension in the tendon (vector T), we need to balance the forces horizontally and vertically.
Horizontally:
Sum of horizontal forces = T - F - w⋅sin(θ) = 0
Vertically:
Sum of vertical forces = w⋅cos(θ) - T = 0
Solving these equations simultaneously, we can find the value of T.
From the equation T - F - w⋅sin(θ) = 0, we can isolate T:
T = F + w⋅sin(θ)
Substituting the given values:
T = 11.3 N + (31.0 N)⋅sin(40.0°)
Calculating T:
T = 11.3 N + (31.0 N)⋅0.6428
T = 11.3 N + 19.93 N
T ≈ 31.23 N
Therefore, the magnitude of vector T is approximately 31.23 N.