Objectives:
- Define torque
- Define static equilibrium
- Determine the resultant of two or more torques
- Determine if an object is in static equilibrium when the forces and torques acting on the object are known
- Determine an unknown force or torque acting on an object, if all the other forces and torques acting on the object are known and the object is in static equilibrium
- Define center of gravity
- Estimate the location of the center of gravity of an object or body
Torque: The turning effect produced by a force
Book example
Ruler example
Lever systems
Inside the body
Mechanical advantage:
Resistance Force/Effort Force
or Effort Arm/Resistance Arm
Tools provide an excellent example of using torques
Torques in diving
Wrestlers use torques to reposition their opponents (full match)
Are there torques in Doug Padilla's race?
Muscle forces and torques
What is the purpose of a patella?
Equilibrium: Forces and Torques Must Balance
What happens as the angle of force changes?
Sample problem
A static analysis of torque about the elbow joint can help estimate the muscle forces required for various arm positions. With the forearm horizontal and holding a 10 kg mass in the hand, determine what muscle force is required to generate a torque balancing the torque created by the weight held in the hand. Determine the muscle force for arm positions which change the angle of the biceps muscle with respect to the forearm from 90 degrees to 75, 60, 45, and 30 degrees. From these calculated muscle forces, sketch a graph of arm angle versus muscle force. What would happen when the angle approaches 0 degrees? Solution  |
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A static analysis of torque about the elbow joint can help estimate the muscle
forces required for various arm positions. With the forearm horizontal and holding a 10 kg mass in the hand, determine what muscle force is required to generate a torque
balancing the torque created by the weight held in the hand. Determine the muscle force for arm positions which change the angle of the biceps muscle with respect to
the forearm from 90 degrees to 75, 60, 45, and 30 degrees.
From these calculated muscle forces, sketch a graph of arm angle versus muscle force. What would happen when the angle approaches 0 degrees?
Force (Weight) of the mass in the hand = m g = (10 kg)(10 m/s/s)
= 100 N
Acting at 40 cm (0.4 m) from the elbow joint center, this weight
creates a torque of:
T = (100 N)(0.4 m) = 40 Nm
With the arm statically holding the weight, muscle force will create
a torque which balances the torque due to the weight in the hand.
Torque due to the muscle force depends on the angle of the muscle
with respect to the forearm:
T = (F muscle)(0.04 m)(sin angle)
If the torques balance, then:
(F muscle)(0.04 m)(sin angle) = 40 Nm
(F muscle) = (40 Nm) / [(0.04 m)(sin angle)]
= (1000 N) / (sin angle)
at 90 degrees: = 1000 N
at 75 degrees: = 1035 N
at 60 degrees: = 1155 N
at 45 degrees: = 1414 N
at 30 degrees: = 2000 N
Thus you can see that as the elbow extends and the angle of the
muscle force with respect to the forearm gets smaller, the force
requirements from the muscle increase dramatically.
As the angle goes toward zero, the muscle force ---> infinitly large!
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Is there a benefit to having the joint position that maximizes the effort arm being a different position than the joint position that maximizes muscle force?
Where is our
center of mass?
Center of mass in a
gymnast
How can we use these principles in
sport?
Side Leap
Jump Height Experiment
Stability: The capacity of an object to return to equilibrium or to its original position after being displaced
Factors affecting stability:
- Height of the center of mass
- Size of base of support
- Weight of object
How do we calculate center of mass position?
Body segment parameters
Body Segment Parameters:
The following tables are excerpted from Plagenhoef et al., 1983. They describe body segment mass as a proportion of total body mass and the location of each segment's center of mass as a proportion of segment length.
| Segment Mass Percents: |
| Segment |
Males |
Females |
| Head & Neck |
8.96 |
8.20 |
| Trunk |
46.84 |
45.00 |
| Upper Arm |
3.25 |
2.90 |
| Forearm |
1.87 |
1.57 |
| Hand |
0.65 |
0.50 |
| Thigh |
10.50 |
11.75 |
| Shank |
4.75 |
5.35 |
| Foot |
1.43 |
1.33 |
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| Segment Length Percents (from proximal): |
| Segment |
Males |
Females |
| Head & Neck |
55.0 |
55.0 |
| Trunk* |
50.0* |
50.0* |
| Upper Arm |
43.6 |
45.8 |
| Forearm |
43.0 |
43.4 |
| Hand |
46.8 |
46.8 |
| Thigh |
43.3 |
42.8 |
| Shank |
43.4 |
41.9 |
| Foot |
50.0 |
50.0 |
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* Estimates for trunk are from Winter (1990) and are calculated from Dempster data (1955) but adapted for the torso alone (without including head & neck). Segment endpoints are from mid-shoulders to mid-hips.
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Calculating center of mass in a body
The shoulder of a tennis player is 1.60m above the ground while the elbow is 1.77m above the ground. Where is the center of mass of the upper arm (the length percent for the upper arm is 43.6%)?
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Dr. Hunter's daughter scores a goal and while celebrating has her left leg in the following position from a side view in meters:
Hip (1.22, 1.01)
Knee (0.90, 0.52)
Ankle (1.23, 0.29)
Heel (1,25, 0.22)
Toe (1.00, 0.10)
Where is the center of mass of the leg if the body segment parameters are as follows:
| Segment |
Length Percent |
Mass (kg) |
| Thigh |
43.3 |
6.83 |
| Shank |
43.4 |
3.24 |
| Foot |
50.0 |
0.98 |
Solution
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High jump example of center of mass