Energy, Work and Power
References
Physics 5th Edition Giancoli, Chapter 6 – Work and Energy
The
Physiological Basis for Exercise and Sport
5th edition – Fox, Bowers and Foss Brown and Benchmark.
Chapter 4 - Measurement of Energy, Work and Power
Contents
Kinetic energy and
the work energy principle
Direct measurement
of Energy: heat production
Energy equivalents
of food and alcohol
The Caloric
Equivalence of Oxygen
Example -
Measuring work on a Cycle Ergometer
Example -
Measuring work on a treadmill
Tutorial Problems
1. When 3.6 litres of O2 are consumed during exercise, how much work, expressed in kJ and kcal, is performed?
2. If the exercise were performed in 1 minute, what is the power output in watts (W)?
3. What proportion of the available energy in an average sized slice of white bread would the exerciser have used in performing the work in the above situation?
Ergometry
To study the energy requirements of various physical activities, we need suitable measurement devices called ergometers.
Ergo = work
Meter = measure
Needs to be reliable, so that if a measured physiological response is different before and after an exercise program, these differences can be attributed to something other than the ergometer itself.
Treadmill
A motor driven conveyor with an adjustable speed and elavation.
Stationary cycle
Provides a resistance against pedalling by
1. Mechanical changing of the resistance experienced by the flywheel
2. Altering the electrical resistance
Other
Tethered swimming, rowing
Energy
We have everyday expressions:
“I haven’t a bit of energy left” or
“He is bursting at the seams with energy”
Energy is a complex concept – for now define it as “the ability to do work” – not strictly valid eg energy associated with heat is often not available to do work.
Energy has units of Joules or calories (J, kJ and cal, kcal).
4.18 joule = 1 cal
There are 6 forms of energy:
1. mechanical
2. heat
3. light
4. chemical
5. electrical
6. nuclear
Each can be converted from one from into another.
An exercising person is converting chemical energy (food) to mechanical energy and heat. (see Appendix Physics in Action)
Our bodies also convert chemical energy into electrical energy – the stimulus to contract a muscle. The source of the energy comes from either breaking up a molecule or putting it back together.
Energy
is conserved and it is a scalar.
If hydrogen reacts with oxygen to produce water and 68.4 kcal, it will require exactly 68.4kcal to pull the water molecule apart again.
Work
In physics work has a very restricted meaning
Work is defined as the product of the magnitude of the displacement times the component of the force parallel to the displacement.
W = Fparallel ´ d or W = Fd cos q
for q being the angle between the force and the displacement.
Work is a scalar quantity – it only has magnitude.
In the above example, the person is doing
work = Fd cos q
§ What if d=0?
§ What if F=0?
§ What if q=90° and F<mg?
A force can be exerted on an object but do no work.
Eg holding a bag of groceries, you do no work – but you get tired!!
Also, you do no work on the groceries if you walk horizontally across the floor at constant velocity.
The upward force is perpendicular to the displacement – hence W=0.
q What is going on?
q You consume energy
q Your muscles get hot
q Your efficiency = 0!
q You are not doing “useful work”
Kinetic energy and the work energy principle
A moving object can do work on another object it strikes.
q A hammer does work on the nail it strikes
q A flying cannonball does work on a wall it knocks down.
q Our moving hand does work by lifting a book.
The energy of motion is called kinetic energy
Translational kinetic energy of an object is
KE= ½ mv2
The net work done on an object is equal to
the change in its kinetic energy.
Wnet = DKE
Work-energy principle
Example
A 145g cricket ball is thrown with a speed of 25m/s.
What is its kinetic energy?
KE = ½ (mv2) = ½(0.145kg)(25m/s)2 = 45J
How much work is done on the ball – assuming it started from rest?
Wnet = DKE = 45J (KE increases and work is positive)
Power
Power is define as the rate at which work is done.
If a 1kg book were raised 1m in 1 second,
the power would be (mg´d) 10kg m per second = 10W
It is important to see the distinction between energy and power.
A person is limited in the work he can do, not only by the total energy required, but also by how fast his energy is transformed – that is by power.
For example, a person may e able to run many kilometres or climbing lots of stairs before having to stop because so much energy has been expended.
On the other hand, a person who runs very quickly up the stair may fall exhausted after only a flight or two. He is limited in this case by power.
Example.
A 70kg jogger runs up a long flight of stairs reaching a height of 4.5m in 4.0s.
q Estimate the jogger’s power output in watts and horsepower.
The work done is against gravity – so that he needs to exert a force of F=mg through a distance y, so
Work W = mgy
The average power output was;
Since there are 746W in one horsepower, the jogger is doing work at just over 1hp – a human cannot work at this rate for very long.
q How much energy did this require?
The
energy required was 
Note, the person needed to transform more energy than this from foodstuffs since some of the transformed energy goes to heat – a person is not 100% efficient – in fact no engine is!
Direct measurement of Energy: heat production
The body metabolised food and some of the resulting energy produces heat and some is used by the body to do work.
This is the First Law of Thermodynamics.
In the later part of the 1800, Max Rubner built an insulated chamber with a circulating water blanket.
A dog was placed inside the chamber called a bomb calorimeter.
Metabolism was indirectly measured by measuring the consumed by the dog in breaking down the foodstuffs.
Heat produced by the dog was measured by noting the change in water temperature.
Results of these early experiments demonstrated unequivocally that energy produced through metabolism of foodstuffs is equal to heat produced in the body – assuming the body does no work.
Equivalently, we can determine the energy or caloric value of food item eg potato, we can put the potato inside a bomb calorimeter and burn it - and measure the temperature increase of the surrounding water.
This is called the direct method of measuring the caloric value.
If, however we measure the oxygen consumption required in metabolising food – this is called the indirect method.
Energy equivalents of food and alcohol
|
Food |
Energy Bomb |
Energy Physiological |
O2 Kcal/liter |
CO2 Kcal/liter |
R |
|
Carbohydrate |
4.10 |
4.02 |
5.05 |
5.05 |
1.0 |
|
Protein |
5.65 |
4.2 |
4.46 |
5.57 |
0.80 |
|
Fat |
9.45 |
8.98 |
4.47 |
6.67 |
0.71 |
|
Alcohol |
7.1 |
7.00 |
7.25 |
0.67 |
0.98 |
|
Mixed diet |
|
|
8.83 |
5.89 |
0.82 |
1. The small differences between the first two columns are due to digestion and some protein losses in the urine.
2. More energy is released from fats as from carbohydrates per gram:
Energy is released when water is formed by combining H2 and O. There
are more hydrogen atoms per oxygen atom in fat than carbohydrates eg
Typical fat, palmitic acid C6H3202
Typical carbohydrate C6H12O6
3. All proteins contain nitrogen in addition to C, H and O. Protein has a low caloric value because this nitrogen must be excreted in the urine and a portion of the energy is lost.
Question: If we have been told that the quantity of oxygen consumed is a measure of the caloric value of the food, how do we know which foodstuff is being metabolised?
A person exercises on a mixed diet, and it is also the CO2 that is produced that permits us to know the mixture of metabolised foods.
The Caloric Equivalence of Oxygen
The respiratory exchange ratio R is
This depends on the relative amounts of Carbohydrates and fats which are metabolised eg
The R-Table
|
R |
kcal per liter of oxygen |
%Carbohydrates |
%Fat |
|
0.7 |
4.686 |
0.0 |
100.0 |
|
0.74 |
4.727 |
12.00 |
88.00 |
|
0.80 |
4.801 |
33.40 |
66.60 |
|
0.85 |
4.862 |
50.70 |
49.30 |
|
0.90 |
4.924 |
67.50 |
32.50 |
|
1.00 |
5.047 |
100.0 |
0.00 |
Measurement of the R- value
Using a manual Douglas Bag, the exhaled air is collected in a bag for later analysis - chemically.
Now there are on-line electronic sensors that can measure the oxygen and carbon dioxide content.
Computation of efficiency
% efficiency is defined as
|
Exercise activity |
%EFF Male |
%EFF female |
|
Horizontal walking |
19.6-35.2 |
|
|
Inclined walking |
20.6-43.0 |
|
|
Swimming |
2.9-7.4 |
2.7-9.4 |
|
Rowing |
10-20 |
|
|
Ice-skating |
11.0 |
|
|
Cycling |
24.4-34.4 |
|
Structural factors such as total body mass, distribution of body mass, distances of fixings of muscles from joints, variations in muscle fibre orientation and length.
A speed of working exists at which the energy required to walk a given distance is minimised.
Example - Measuring work on a Cycle Ergometer
One complete turn of the pedals moves a point on the rim 6m.
The braking force is calibrated in N.
A patient pedals the cycle for 10 minutes at a rate of 50rpm with a force resistance of 3N. The R-value is measured to be 0.85.Oxygen is consumed at a rate of 2l/min.
q What is the total work output?
Wout = F ´ d = 3N ´ 50 rev/min ´ 10 min ´ 6 m/rev
= 3N ´ 3000m = 9000 J = 9kJ
But 1cal = 4.186J so 1kcal=4.186kJ
So, work output = Wout = 2.15kcal
For clarity, lets just repeat the question:
A patient pedals the cycle for 10 minutes at a rate of 50rpm with a force resistance of 3N. The R-value is measured to be 0.85.Oxygen is consumed at a rate of 2l/min.
q What is the work input?
From the R-table, at R=0.85, 4.865 kcal energy is consumed per litre of oxygen.
Total VO2 = 2.0 l/min ´ 10 min = 20 l
Total calories consumed is thus
Win = 20l ´ 4.865 kcal/l = 97.3 kcal
q What is the %EFF?
Often, cycle ergometers are calibrated to give a direct readout in power i.e. watts.
Efficiency may still be calculated so long as the numerator and denominator are expressed in identical units.
Example - Measuring work on a treadmill
If a patient is walking or running on a horizontal treadmill, he would not be performing any useful work – and hence his efficiency would be 0!!
The patient would get tired, but according to the physicist, he would not be doing any work.
All the energy the patient is metabolising is degraded as heat. The same holds true if you hold a book at arms length.
Usually the treadmill gradient is reported as a %grade instead of degrees.
% grade = rise in m per 100m of run
% grade = tan q ´ 100 e.g. 8° = 0.1405 ´ 100 = 14.05%.
A 73kg patent runs at 4.8km/hr for 30 minutes on an 5° treadmill and consumes oxygen at a rate of 1.2l/min at R=0.80.
q What is the output work?
The patient is applying a force equal to his own weight along a distance equal to the vertical height travelled.
To think about this correctly, consider the patient running along a road having an incline of 8°.
The total distance travelled is 4.8km/hr ´ 0.5hr = 2.4km.
The vertical height travelled, y
y = 2.4km ´ sin (5°) = 0.21km = 210m.
Work = 73kg ´ 9.8ms-2 ´ 210m
= 1.5 ´105J / 4.18 J/cal = 3.59´104cal = 35.9kcal
q What is the energy input?
With R = 0.82, 1 litre O2 = 4.8 kcal
So, total volume of O2 consumed = 1.2 l/min ´ 30 min = 36liter
So total energy used = 36 l ´ 4.8 kcal/l = 173 kcal
q What is the efficiency?
%EFF = 35.9/173 = 20.8%