Which Exercise Requires More Abdominal Effort: Pilates or Sit-Ups?

[mahkum]

This question requires you to model the human body so that you can analyse the static loads, (comparing the pilates position and the abdominal sit-up position, figure above)
Using the body mass ‘M’ of 94 kg and a body height ‘h’ of 1.73 metres, establish which exercise requires the most effort from the abdominal muscles to hold the position shown and by what factor.
The table shows the details of body parts so that comparison between the two positions can be performed.

Table:
Body Parts Mass in terms of M,
the person’s mass Length, I, in terms of h, the person’s height Position of cetnre of mass along from upper end of the body part
Thighs 0.2M 0.24h 0.43/
Shanks + feet 0.13M 0.29h 0.47/
Head + neck 0.08M 0.182h 0.36/
Trunk + arms 0.59M 0.332h 0.50/

For the purpose of this question, the mass and position of the hands and arms can be considered to be part of the trunk. Place your origin at the join between the bottom of the trunk and the top of the thighs, i.e. where the abdominal muscles act. Assume that the angle the legs in the pilates position makes with the floor is 60o , whereas the angle the head and neck make with the floor is 45o.
For the sit-up exercise, the trunk is at 30o to the floor and the legs are bent so that the knee joint makes a 90o bend.
To gain full marks:
a) Give valid assumptions on how you model the body components
b) Sketch clear, fully labelled free-body diagrams of the two positions, showing the known and unknown external acting forces and moments
c) Provide accurate equilibrium equations by resolving the loads and moments
d) Calculate the unknown external loads and moments
e) Calculate the moment provided by the abdominal muscles to achieve the position

Homework Equations

The Attempt at a Solution

 

[Valkarie]

a) Assumptions:• The human body is modelled as a collection of rigid body parts. • The centres of mass for each body part are assumed to be at the midpoint of their length. • The position of the centre of mass is taken to be along the longitudinal axis of the body part. • The external forces and moments acting on the body are assumed to be constant. • The moment is calculated as the sum of the moments around the origin. b) Free-body diagrams:Pilates Position:Abdominal Sit-up Position:c) Equilibrium equations:Pilates Position:Sum of Forces in x-direction = 0Fx = 0Sum of Forces in y-direction = 0Fy = 0Sum of Moments around origin = 0M = 0Abdominal Sit-up Position:Sum of Forces in x-direction = 0Fx = 0Sum of Forces in y-direction = 0Fy = 0Sum of Moments around origin = 0M = 0d) Calculating the unknown external loads and moments:Pilates Position:Fx = 0 = M*sin(60o) + W*cos(60o)M = W*cos(60o)/sin(60o)Fy = 0 = M*cos(60o) - W*sin(60o)W = M*cos(60o)/sin(60o)M = W*sin(60o)/cos(60o)Abdominal Sit-up Position:Fx = 0 = M*sin(30o) + W*cos(30o)M = W*cos(30o)/sin(30o)Fy = 0 = M*cos(30o) - W*sin(30o)W = M*cos(30o)/sin(30o)M = W*sin(30o)/cos(30o)e) Calcul