A person bending forward to lift a load with his back, as shown in the figure, rather than with his knees can be injured by large forces exerted on the muscles and vertebrae. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, and to understand why back problems are common among humans, consider the model shown in the figure of a person bending forward to lift a 140 N object. The spine and upper body are represented as a uniform horizontal rod of weight 380 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is 12.0°. Find the tension in the back muscle (T).
To find the tension in the back muscle (T), we will analyze the forces acting on the system.
1. Identify the forces acting on the system:
- Weight of the object being lifted (F_object) = 140 N, acting downward
- Weight of the upper body and spine (F_weight) = 380 N, acting downward
- Tension in the back muscle (T), acting upward at a 12.0° angle
2. Determine the net torque about the pivot point (base of the spine) to find the tension in the back muscle.
- The net torque should be zero for the system to be in rotational equilibrium.
3. Calculate the torque due to the weight of the object (τ_object):
- The distance between the pivot point and the object's weight is unknown, so let's call it 'D_object'.
- The torque due to the weight of the object is τ_object = F_object * D_object * sin(90°) = F_object * D_object.
4. Calculate the torque due to the weight of the upper body and spine (τ_weight):
- The distance between the pivot point and the center of mass of the upper body and spine is unknown, so let's call it 'D_weight'.
- The torque due to the weight of the upper body and spine is τ_weight = F_weight * D_weight * sin(90°) = F_weight * D_weight.
5. Calculate the torque due to the tension in the back muscle (τ_T):
- The distance between the pivot point and the point of attachment of the back muscle is two-thirds of the way up the spine.
- Let's call the length of the spine L. The distance to the point of attachment is D_T = 2/3 * L.
- The torque due to the tension in the back muscle is τ_T = T * D_T * sin(12.0°) = T * D_T * sin(12.0°).
6. Set up the torque equation:
The net torque should be zero, so the sum of the torques equals zero.
τ_net = τ_object + τ_weight + τ_T = 0
Therefore,
F_object * D_object + F_weight * D_weight + T * D_T * sin(12.0°) = 0
7. Solve for the tension in the back muscle (T):
T * D_T * sin(12.0°) = -F_object * D_object - F_weight * D_weight
Finally, you can calculate the tension in the back muscle (T) using the values given in the problem statement. Make sure to substitute the appropriate lengths and angles into the equation and solve for T.
Note: The actual calculation requires knowing the specific values for L, D_object, D_weight, F_object, and F_weight.