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 compressional force in the spine. (Rx).

Question

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 compressional force in the spine. (Rx).

Answer 1

To find the tension in the compressional force in the spine, we can first analyze the forces acting on the system and then apply the principles of equilibrium.

Let's break down the forces acting on the system:

1. Weight of the upper body: The weight of the upper body is given as 380 N, acting downwards at the center of mass of the uniform horizontal rod.

2. Weight of the object being lifted: The weight of the object being lifted is given as 140 N, acting downwards.

3. Tension in the erector spinalis muscle: The tension in the erector spinalis muscle acts upwards at an angle of 12.0° with the spine.

4. Tension in the compressional force in the spine: This is the unknown force we need to find (Rx).

Since the system is in equilibrium (not moving), the sum of the forces in the vertical direction should be zero.

Let's calculate the vertical components of the forces:

Vertical component of the weight of the upper body = 380 N * sin(90°) = 380 N
Vertical component of the weight of the object = 140 N * sin(90°) = 140 N
Vertical component of the tension in the erector spinalis muscle = Tension * sin(12.0°)

According to the principle of equilibrium, the sum of the vertical components of these forces must be equal to zero:

380 N + 140 N + Tension * sin(12.0°) = 0

Now we can solve this equation to find the tension (Rx):

Tension * sin(12.0°) = -380 N - 140 N
Tension = (-380 N - 140 N) / sin(12.0°)

Calculate the tension using the given values and the equation above to find the tension in the compressional force in the spine.