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 can start by analyzing the forces acting on the system.
1. Weight of the upper body: The weight of the upper body is given as 380 N, acting downward at the center of mass of the horizontal rod.
2. Weight of the load: The weight of the load is given as 140 N, acting downward at the location of the load.
3. Tension in the back muscle: The tension in the back muscle (T) acts upward at the attachment point on the spine.
4. Reaction force at the pivot: There is a reaction force at the base of the spine that counteracts the sum of all the other forces.
Now, let's break down the forces and set up equations to solve for T.
Considering the vertical forces:
Tension in the back muscle (T) - Weight of the upper body (380 N) - Weight of the load (140 N) - Reaction force = 0
Considering the torques:
The torque due to the weight of the upper body about the pivot = (380 N) * (distance from the pivot to the center of mass)
The torque due to the weight of the load about the pivot = (140 N) * (distance from the pivot to the load)
The torque due to the tension in the back muscle about the pivot = T * (distance from the pivot to the attachment point)
Since the system is in equilibrium, the sum of all the torques must be zero.
Now, we need to find the distances from the pivot to the center of mass and the pivot to the attachment point. Unfortunately, the figure is not provided, so we cannot determine these distances numerically.
Therefore, without specific values for the distances, we cannot determine the exact value of the tension in the back muscle (T).