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
where upper body weight is 430N
lower body weight is 370N
upper body mass moment arm is 23cm
lower ody mass m0ment arem is 15cm
weight mament arem is 35cm
muscle force moment ARM IS 50CM
The ground reaction force (GRF) is the force exerted by the ground on the body in response to the body's weight and muscle forces. The position and magnitude of the GRF can be calculated using the following equation:
GRF = (Upper Body Weight + Lower Body Weight + Muscle Force) x Moment Arm
GRF = (430N + 370N + 430N) x 50cm
GRF = 1,230N x 50cm
GRF = 61,500Nm
The GRF is 61,500Nm and is located at the mid-point between the fourth and fifth lumbar vertebrae.
To calculate the position and magnitude of the ground reaction force, we will need to analyze the forces acting on the weight-lifter and apply equilibrium conditions.
1. Start by calculating the moment exerted by the erector spinae muscles:
Moment = Muscle force x Muscle force moment arm
Moment = F x r
Moment = F x 50 cm
2. Set up the equilibrium equation:
Sum of clockwise moments = Sum of anticlockwise moments
Clockwise moments:
Moment exerted by erector spinae muscles = F x 50 cm
Anticlockwise moments:
Moment exerted by upper body weight = Upper body weight x Upper body mass moment arm
Moment exerted by lower body weight = Lower body weight x Lower body mass moment arm
Moment exerted by weight lifted = (Weight lifted - Lower body weight) x Weight moment arm
Summing these moments, we get:
F x 50 cm = (Upper body weight x Upper body mass moment arm) + (Lower body weight x Lower body mass moment arm) + ((Weight lifted - Lower body weight) x Weight moment arm)
3. Substitute the given values into the equation:
F x 50 cm = (430 N x 23 cm) + (370 N x 15 cm) + ((Weight lifted - 370 N) x 35 cm)
4. Solve for F:
F = [(430 N x 23 cm) + (370 N x 15 cm) + ((Weight lifted - 370 N) x 35 cm)] / 50 cm
Substitute the value for the weight lifted. In this case, the weight lifted is the same as the upper body weight since it is equal to 430 N:
F = [(430 N x 23 cm) + (370 N x 15 cm) + ((430 N - 370 N) x 35 cm)] / 50 cm
Simplify the equation:
F = [9890 cm N + 5550 cm N + (60 N x 35 cm)] / 50 cm
F = [15440 cm N] / 50 cm
F = 308.8 N
Therefore, the magnitude of the ground reaction force is approximately 308.8 N.
To find the position of the ground reaction force, we need to determine the horizontal and vertical components of the force.
1. The horizontal component of the ground reaction force is equal to the horizontal component of the weight lifted, which can be calculated as:
Horizontal force component = Weight lifted x cos(40 degrees)
Substituting the value for the weight lifted:
Horizontal force component = 430 N x cos(40 degrees)
2. The vertical component of the ground reaction force is equal to the sum of the vertical components of the upper body weight and the lower body weight, which can be calculated as:
Vertical force component = (Upper body weight + Lower body weight) - Weight lifted x sin(40 degrees)
Substituting the given values:
Vertical force component = (430 N + 370 N) - 430 N x sin(40 degrees)
Therefore, the position of the ground reaction force is determined by the horizontal and vertical components calculated above.
To calculate the position and magnitude of the ground reaction force, we need to consider the forces acting in equilibrium.
First, let's calculate the upper body moment caused by the weight of the upper body:
Upper body moment = upper body weight * upper body mass moment arm
= 430 N * (23 cm / 100 cm)
= 99.1 Nm
Similarly, let's calculate the lower body moment caused by the weight of the lower body:
Lower body moment = lower body weight * lower body mass moment arm
= 370 N * (15 cm / 100 cm)
= 55.5 Nm
Now, let's analyze the forces acting on the body. We have the ground reaction force acting vertically upward, and the muscle force acting at an angle of 40 degrees to the vertical. These two forces should balance the moments to keep the body in equilibrium.
The equation for the moments in equilibrium is:
Lower body moment + Upper body moment = Muscle force moment + Weight moment
Rearranging this equation, we get:
Muscle force moment = Lower body moment + Upper body moment - Weight moment
Plugging in the given values:
Muscle force moment = 55.5 Nm + 99.1 Nm - (430 N * (35 cm / 100 cm))
= 55.5 Nm + 99.1 Nm - 150.5 Nm
= 4.1 Nm
Now, let's calculate the magnitude of the ground reaction force:
By taking the moment arm of the ground reaction force as the distance between the fourth and fifth lumbar vertebrae (50.0 mm or 5.0 cm), we can use the equation:
Muscle force moment = Ground reaction force * Ground reaction force moment arm
5.0 cm * Ground reaction force = 4.1 Nm
Ground reaction force = 4.1 Nm / 5.0 cm
= 0.82 N
Therefore, the magnitude of the ground reaction force is 0.82 N.
To find the position of the ground reaction force, we need to calculate the horizontal distance from the fourth and fifth lumbar vertebrae. Since the weight-lifter is in equilibrium, this distance should be the same as the combined weight moment (weight x weight moment arm) divided by the magnitude of the ground reaction force:
Horizontal distance = (430 N * (35 cm / 100 cm)) / 0.82 N
= 185 cm
Therefore, the position of the ground reaction force is located 185 cm horizontally away from the mid-point between the fourth and fifth lumbar vertebrae.