Volume 3, Physics 208
Last updated on May 25, 1999
The muscle can deliver the largest force at zero velocity of the muscle contraction. The muscle can contract with a certain muscle speed which is dependent of the length of the muscle. Typically, the muscle can contract at between 2 and 4 times the length of the muscle per second. Both of these emperical observations results from the microscopic muscle contraction process.
There are molecular motor in the muscle which draw micro sections of the muscle together. These motors have a intrinisc maximum speed. This speed is approximately: . The higher speeds observed for macroscopic muscles [groups of these fibers] is the result of the end-to end joing of the fibers producing N times the speed of contraction per fiber where N is the number of fibers joined together end-to-end.
The molecular motors must have traction to draw the muscle substructures together for a contraction. The efficiency of this traciton is highest for zero or low speeds but deteorates with higher speed. This slippage of the molecular motors results in lower forces as the muscle contraction speed is increased.
Recall that the power produced by a muscle is P = F * velocity. At zero velocity, no power is produced and at the highest velocity, F=0 and no power is produced. For the functional shape of the force-speed curve, the maximum power should occur at about 1/3 maximum speed. This is born out by the maximum power production of cyclists at about 100 rpm. Cyclists, in training, can obtain 300 rpm with no load which makes the maximum power output at about 1/3 max speed. A cyclist can continually adjust the gears on the bike so that he can be pedalling at the optimum muscle speed independent of the conditions [hill, wind].
Figure showing the force/power vs muscle speed. The ordinate is in units of maximum force and the abcissa is in units of the maximum speed for the muscle. The secondary ordinate is in units of power scaled by maximum force and maximum velocity. Peak power occurs at about 1/3 maximum velocity with a power of 1/10. The muscle can develop the highest forces when it is being stretched.[negative velocity].
In everyday language, we sometimes use power and energy interchangeably. Here we will be quite strict about proper useage. Power is the time rate of work.
Power = work/time
watt = joule/sec
If the athlete lifts the 100 kg barbell by 2 m and this is done in one second, the power is P = work/time = 2000 J/1 sec = 2000 watts.
The Jump Reach is a standard test used by pro scouts to evaluate athletes. Basically the Jump Reach is a stationary vertical jump in which the athlete tries to touch a scale on the wall as high as possible. What is measured is the height of the jump, but what is inferred is the power output. What follows is an analysis of a Jump Reach by a UW football player.
The athlete remained in the air for 0.76 seconds which implies a jump reach height of 0.72 m. The time from the swing of the arms until the feet leave the floor is a total of 0.33 seconds. The final thrust of the legs takes about 0.13 seconds and the leg thrust raises the body from the squat position to a standing position, a height of about 0.3 m. The complete jump then has raised the CM of the athlete by a height of 1.02 m. The mass of the athlete is about 100 kg. If most of the energy comes from the leg thrust, the power that must be developed is:
Average Power = Energy/(time of the leg thrust) = mgh/(time of the leg thrust)
P = 100*9.8 * 1.02/0.13 = 7700 watts. This is a lot compared with the power exhibited during the stair climb of the class members (which was about 800 watts on the average). As the athlete moves faster near the end of the jump, the power requirements are even higher. Max Power = 2*(average power) = 15400 watts!! [assuming a constant acceleration]
If we average the power over the complete cycle of arm swing and torso thrust, the power required is much less.
Max Power = 2*100*9.8*1.02/0.33 = 6060 watts which is still quite high.
The important power effects are probably in the 0.2 second interval when the body and arms begin their upward journey. This makes the power average to be 5000 watts with a maximum power required of about 10,000 watts which is a lot! Some of the 140 kg linemen can jump this high so that they produce power proportionately higher.
Power from Muscle Mass
The power produced by a muscle is dependent on its mass. Recall the law of muscles: Force depends on cross sectional area. The distance of the application of force depends on the length. Since work = force*length, the work that is possible with a muscle depends on the volume or mass. The power that is possible depends on the speed of the muscle movement. A high speed means a short time for the movement to yield high power.
To support that the measured relation of F~cross sectional area of muscles that is measured in isolated muscles, we examine the performance of championship weight lifters. We examine the relation of sum of the weights obtained in the snatch and the jerk in Olympic competition. [records in 1987].
These data show world record weight lifts as a function of the weight of the athlete. Note that when the athlete's weight doubles, the lift does NOT double. The force produced does not grow linearly. If the force grows as the cross sectional area, this means that force varies as L2. Since the weight of a person varies as the volume or L3, the force that can be produced should vary as weight2/3.
F ~ L2 and weight ~ L3 or L ~ weight1/3
F ~ weight2/3
To show this functional relation, we can divide the lift weight by athlete's weight and then also by weight2/3. This is shown below:
If the hypothesis fits, the ratio should be a horizontal line. The hypothesis that F ~ L2 fits pretty well while the F ~ L3 does not fit. Clearly the law of muscles works in actual sports performance in a direct measurement.
One author asserts that the maximum power that one pound of muscle can extract is 1/8 horsepower. 1 horsepower = 750 watts and 1 pound has 0.45 kg of mass.
The 1/8 HP per pound of muscle is a conservative estimate. There is quite a lot of variation in the muscle performance. Here's a summary from the literature:
There is one complication; the maximum force that a muscle can exert depends on the speed of the contraction. The highest force can be produced with zero speed. At the maximum speed, no force is produced. Studies have shown that the maximum power is obtained if the muscle is contracted at about 1/3 of the maximum muscle speed.
Taking these factors into account, the power per pound of muscle can be estimated to be ~1/4 HP for the average performance of the above factors.
Specific power of muscles: ~400 watts/Kg. This may vary by about 50%.
Let's apply this knowledge to the jump reach analyzed earlier.
For the jump reach performance analyzed above, the power required if only the legs were thrusting was 15,400 watts. The mass of muscle required for this would be 15,400/400 = 38 kg. To appreciate how much this is, imagine a stack of steaks that weighs 84 pounds! Compare this to the muscles in the quadraceps and gluteus maximus. There isn't enough mass in just those muscles.
It's clear that the jumping performance requires using more muscles of the body than just the leg muscles. The arm swing and torso thrust are important, so the average power is somewhat less than 15,400 watts. Assume that the power is about 10,000 watts, then the required amount of muscle is about 54 pounds. This much muscle can easily be accomodated between the torso, arms, and legs. [A top athlete will have about 1/2 his or her weight in muscles.]
A grasshopper, a cat, a human and a horse can all jump about 1 m. That is, they can raise their CM by about 1 m. Intuitively, we think that the scaling law for jumping should be that a jump is proportional to the size of the jumper. If this were true, then a grasshopper jumping to a height of about 100X his length would be equivalent to a human jumping 200 m up! This is the basis of Spiderman's super-hero status: he supposedly has the proportionate strength and speed of a spider. Is this really how the scaling works?
We can think about how size affects jumping performance using the Law of Muscles. Suppose the characteristic size of a person is height = a*L, width = b*L, and depth = c*L, where L is the characteristic length that scales the person.
Force is proportional to area or L2 since the force is proportional to the cross sectional area of the muscle.
Mass is proportional to volume or L3
Acceleration = F/m (Newton's 2nd Law) and so is proportional to L2/L3 = 1/L
The conclusion is that the smaller the person (or animal), the greater the acceleration that is possible. In athletic events where acceleration is important the athletes are usually of a small stature. Gymnastics, diving, and ice-skating are examples of events where quick takeoffs and spins favour those who can develop large accelerations, i.e.those with small size.
Does this mean that the smaller person can jump higher or would have larger velocity? NO. To obtain velocity, we must accelerate over a period of time. The smaller the person, the smaller the distance (and time) available for the acceleration.
v2 = 2ax, where x is proportional to L.
We then have v2 is proportional to (1/L * L) = 1. This means that speed is independent of size. Note that jumping height is proportional to take-off speed. The jump reach is independent of size for athletes. Of course, it depends on strength, but not on height. This is why species with quite different sizes have about the same jump height or take off velocity. It also means that if the proportion is taken properly then the proportionate strength of a spider is the same as that of a man, so Spiderman would be just a normal Joe.
The shortest homerun ball requires a launch velocity of 45 m/s and a bat velocity of 24 m/s when striking a 45 m/s fastball with a 30 oz bat. The kinetic energy of the 5.1 oz ball is Kball = 1/2*mv2 = 1/2*0.14*452 = 147 J. The kinetic energy of the bat before the collision is: Kbat = 1/2*0.85*242 = 245 J. Studies of batters have shown that they bring the bat around to hitting position in about 0.13 seconds. The average power to produce the kinetic energy of the bat is thus Pavg = 245/0.13 = 1884 watts. The peak power will be almost 3800 watts. From our study of power density of muscles, the agonist muscles involved must have a mass of 9.5 kg. Since each agonist must have an antagonist, the sum of the two must be about 18 Kg or 36 pounds of muscle. The muscle mass in the shoulders and arms are insufficient to generate the needed power. The power must be generated with the big muscles of the body. These are the muscles in the torso and hips.
A big question is how does the power generated by the hips get delivered to the bat? The hips move rather slowly in the batting process. To understand how this occurs, we should reflect upon how a whip works. The driver of a horse-drawn rig will use modest forces and velocities but the tip of the whip will attain super-sonic velocities. Other examples of this are the crack the whip with a chain of people or the snapping towel in the locker room. The general idea is that the kinetic energy of the larger mass [moving at small velocities] is transferred to the smaller mass, but the velocity must be much larger to conserve energy.
In the hit, the massive hips and torso are moved with a small velocity but the kinetic energy is large. This energy is delivered to the bat with a resultant high velocity. The process starts with the bat being relatively stationary and the hips and torso move forward leaving the bat behind. As the arms are stretched out, the bat is brought around at the end. It's important that the wrist remains loose so that the energy is efficiently transferred.
The whip-like process is important for the pitcher as well. A study of the pitching motion shows the same process of leaving the ball and hand behind while the body is in forward motion, then whipping the arm for the pitch.
Practice and timing are ways to achieve this whip-like action in hitting and throwing. When you try this, remember that you will always let the ball or the bat lag behind the rest of the motion and it only comes up to speed at the last moment.
1 Kilocalorie = 4186 joules
1 day = 24*60*60 sec = 86400 sec
1 watt = 1 joule/sec
3000 kilocalories/day * 4186 J/kilocalorie * 1 day/86400 s = 143 J/s = 143 watts
The above is the average power output. The peak power can be 100X higher for short time intervals.
Most [75% or more] of the power output is in waste heat so that the maximum power to actually carry out mechanical work is only 360 watts while the body is pumping out 1080 watts of heat. This is the aerobic limit [using oxygen].
For VERY short bursts of power, the body puts out about 10-50X more work with anaerobic muscles [no oxygen needed]. This lasts for less than a minute.
Energy from digested food is stored mainly in the muscles with the excess stored as fat. The reserves stored in the muscles are used first for work. The PC [Phosphocreatine] fuels type II muscles which are called into play for the big forces. Unfortunately, there is only about a 20 second supply of PC in the muscles. Of course, this is all one needs for the 100-200 meter sprints and for jumping and pole vaulting. You can hold your breath and carry out short periods of intense excercise with no adverse consequences.
For extended work, stored glycogen supplies most of the energy. This process produces the waste product Pyruvic acid. At lower power levels, the oxydation process can burn up the Pyruvic acid so that this does not poison your muscles. At higher power levels, the excess Pyruvic acid is converted to lactic acid which causes the burning pain sensation in muscles. When this happens the muscles scream for you to stop. This lactic acid is metabolized when the excercise stops and the oxygen is available to oxidize the lactic acid.
In a competition like a 1 mile race, the idea is to stay within the aerobic limit [not produce lactic acid] for the major part of the race. In the finishing part of the race, you can go all out and burn up your PC and produce lots of lactic acid. Later you can recover.
In rowing, the psychological advantage of being ahead at the beginning of a race is important since the rowers face backwards. In this case, the athletes frequently burn their PC and build up lactic acid at the beginning of the race. The finish is an agony of pain as the lactic acid builds up to ever higher levels. Part of rowing training is involves lactic acid tolerance.
Example: Baseball hitting.
The distance from the pitcher to the batter is 60' = 18 m.
A major league pitch averages about 90 mph = 39 m/sec.
The time of travel is distance/speed = 18/39 = 0.46 s.
A batter must interpret the path of the ball, decide to swing, then bring the bat around to the hitting position within this 0.46 s.
Obviously this is feasible...but what is the limit?
A measured response time shows that it takes ~0.25 sec to respond to external stimuli.
How can you make yourself quicker?
Use your muscles closer to your head. It turns out that the dominant time is the propagation time of the nerve impulse travelling from your head to your muscles. The shorter the nerve length, the faster the response.
There is a limitation in the speed of interpretation of the event to decide to respond. This is somewhat trainable so that you can get faster. But the ultimate limit set by the propagation time cannot be improved. Hence, short people have a slight inherent advantage in response time over tall people.
Assumptions:
The outcome is that the speed, v, varies as n1/9, where n is the number of rowers.
Since about 75% of the energy output of your body goes to heat, and the human body functions well only if it is maintained within a few degrees of its average temperature, we must assure that the body has adequate cooling. This is not a problem for one jump or one swing with the bat, but is a limiting problem for endurance events.
One very important lesson in sports is that the cooling process is absolutely necessary for people, just as it is for automobile engines. An engine will function only for a short time if the cooling process is removed. For athletes in endurance sports, the heating of the body is the most severe limitation for continued performance. Cylists can perform so well because they have a good stream of wind (20-30 mi/hr) to cool their bodies. The evaporation process is the most important part of the cooling system. If your sweat beads up on your body, it is a sign of a failure of the cooling process and signals the limits to which you can push yourself.