When a weight-lifter lifts a weight equal to his upper body weight his trunk makes an angle of 40 degrees, with the vertical, as shown in the diagram below. In this position the erector spinae muscles exert a large force on the sacrum at an angle of 40 degree to the vertical with a moment arm of 50.0 mm about the mid-point between the fourth and fifth lumbar vertebrae. Assuming that the weight-lifter is quasi-static.

(i) Calculate the position and magnitude of the ground reaction force

To calculate the position and magnitude of the ground reaction force, we need to consider the equilibrium of forces acting on the weight-lifter.

First, let's find the weight of the weight-lifter. Since he is lifting a weight equal to his upper body weight, his total weight can be considered as twice his upper body weight. Let's denote it as W.

The weight of the weight-lifter acts vertically downward through his center of mass, which we can assume to be located at the midpoint between the fourth and fifth lumbar vertebrae.

Now, let's analyze the forces involved:

1. Weight of the weight-lifter (W): This force acts vertically downward through the center of mass.

2. Ground Reaction Force: This force acts vertically upward at a certain distance from the center of mass.

3. Force exerted by erector spinae muscles: This force acts at an angle of 40 degrees with the vertical and creates a moment about the mid-point between the fourth and fifth lumbar vertebrae.

According to the principle of equilibrium, the sum of all forces acting on the weight-lifter must be zero, and the sum of all moments about any point must also be zero.

Let's denote the position of the ground reaction force as d from the center of mass.

Using the equilibrium condition, we can set up the following equation:

Vertical Forces: Ground Reaction Force - Weight = 0
Moment equation: Force exerted by erector spinae muscles * moment arm - Weight * d = 0

Substituting the values given in the problem:

1. Vertical Forces: Ground Reaction Force - W = 0
2. Moment equation: Force of erector spinae muscles * 50.0 mm - W * d = 0

Now, we can solve these two equations to find the magnitude and position of the ground reaction force.

(i) Calculate the position and magnitude of the ground reaction force:

1. From the vertical forces equation, we get:
Ground Reaction Force = W

2. Substituting this into the moment equation, we have:
Force of erector spinae muscles * 50.0 mm - W * d = 0

3. Rearranging the equation, we get:
Force of erector spinae muscles * 50.0 mm = W * d

4. Dividing both sides by W:
Force of erector spinae muscles * 50.0 mm / W = d

Now, plug in the values for the force of erector spinae muscles, the moment arm, and the weight of the weight-lifter into this equation to find the position of the ground reaction force.