Physics Class 9/26/11
Answers to questions are due on 10/3/11
A ball travels between two points 10 cm apart in 1/2 second, while accelerating up a ramp on which the acceleration has magnitude 15 cm/s^2, is found to have velocity 20.75 cm/s at the first point and 16.25 cm/s at the second. This can be done using direct reasoning (displacement and time interval give average velocity, acceleration and time interval give change in velocity; the average velocity being halfway between initial and final velocity we can deduce the initial and final velocities from average velocity and change in velocity). The problem can also be solved using the equations of motion.
If the ball is allowed to coast up the ramp until it reaches its furthermost point, the interval of interest will start when the ball passes on of the two original points and ends when the ball comes to rest. We know the ball's velocity at either of the two original points, so whichever we choose as the initial point for the new interval we will know the initial velocity for the interval. We know the acceleration and the ball comes to rest at the final instant so the final velocity is zero.
Choosing the second of the original points as our initial point, and choosing upward as the positive direction we have v0 = 16.25 cm/s, vf = 0 and a = -15 cm/s^2. We easily reason out this situation (average velocity is (v0 + vf) / 2 = 8.1 cm/s, approx.; time interval = change in velocity / acceleration = -16.25 / (-15 cm/s^2) = 1.08 second so displacement = average velocity * time interval = 8.1 cm/s * 1.08 s = 8.7 cm).
We now ask how fast the ball would be traveling when it reaches the lower end of the ramp, which is 10 cm below the first of our original points. For this interval we will choose the initial point to again be the second of our original points, at which the ball was moving at 16.25 cm/s. Assuming some uncertainty in our observations we need to be more modest with our significant figures, so we'll use 16 cm/s as a reasonable approximation to our velocity at that point. The final event of the interval is the ball reaching the end of the ramp. The acceleration is still 15 cm/s^2 down the incline, and our final point (the end of the ramp) is 20 cm below our chosen initial point. Choosing downward as positive, our acceleration is 15 cm/s^2, our displacement is 20 cm and our initial velocity is -16 cm/s. To solve we will use the fourth equation of motion to get the final velocity, obtaining about +-29 cm/s. Since we know the velocity is down the ramp, which is our chosen direction, we conclude that the final velocity is +29 cm/s.
If frictional force is 1% of gravitational force, and that our 60-gram ball in the absence of friction accelerates at 15 cm/s^2, we draw the following conclusions:
In the absence of friction the net accelerating force is the component of the gravitational force parallel to the ramp, which since the acceleration is 15 cm/s^2 gives us
F_parallel = 60 grams * 15 cm/s^2.
Falling freely under the force of gravity alone the ball would accelerate downward at 980 cm/s^2. The force exerted by gravity on the ball is therefore
F_grav = 60 grams * 980 cm/s^2 = 60 000 g cm/s^2, approximately
and 1% of this force is about
F_frict = .01 * 60 000 g cm/s^2 = 600 g cm/s^2.
If the ball is traveling up the incline the frictional force is down the incline, as is the parallel component of the gravitational force, so that F_net is 600 g cm/s^2 + 900 g cm/^2 = 1500 g cm/s^2 down the incline.
If the ball is traveling down the incline the frictional force is up the incline, whilch the parallel component of the gravitational force is still down the incline, so that F_net is 900 g cm/s^2 - 600 g cm/s^2 = 300 g cm/s^2 down the incline.
So if the ball travels up the incline its acceleration is
a_up = F_net_up / m = 1500 g cm/s^2 / (60 g) = 25 cm/s^2 down the incline
and if it travels down the incline its acceleration is
a_down = F_down / m = 300 g cm/s^2 / (60 g) = 5 cm/s^2 down the incline.
These ideas are all relevant to the questions due on 9/28/11.
At the end of this document, just below the questions to be completed by University Physics students, are two interesting articles. Everyone should read through those articles and answer the questions that follow them.
Questions and Assignment (University Physics only):
`q001. For the hotwheels car you observed, what is the force per gram while the car is going up the incline, and while it is going down the incline?
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What do you conclude is the frictional force acting on the car, per gram of car mass?
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What is this result, as a fraction of the gravitational force acting on a gram of mass?
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`q002. For a 4 kg cart on a 7 degree incline, what is the net force acting on the cart, assuming that the frictional force is 10% of the total gravitational force acting on it?
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What therefore would be its acceleration when coasting up the incline, and when coasting down the incline?
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Interesting physics in two articles:
Article from 9/23/11
The physics world is abuzz with news that a group of European physicists plans to announce Friday that it has clocked a burst of subatomic particles known as neutrinos breaking the cosmic speed limit — the speed of light — that was set by Albert Einstein in 1905.
If true, it is a result that would change the world. But that “if” is enormous.
Even before the European physicists had presented their results — in a paper that appeared on the physics Web site arXiv.org on Thursday night and in a seminar at CERN, the European Center for Nuclear Research, on Friday — a chorus of physicists had risen up on blogs and elsewhere arguing that it was way too soon to give up on Einstein and that there was probably some experimental error. Incredible claims require incredible evidence.
“These guys have done their level best, but before throwing Einstein on the bonfire, you would like to see an independent experiment,” said John Ellis, a CERN theorist who has published work on the speeds of the ghostly particles known as neutrinos.
According to scientists familiar with the paper, the neutrinos raced from a particle accelerator at CERN outside Geneva, where they were created, to a cavern underneath Gran Sasso in Italy, a distance of about 450 miles, about 60 nanoseconds faster than it would take a light beam. That amounts to a speed greater than light by about 0.0025 percent (2.5 parts in a hundred thousand).
Even this small deviation would open up the possibility of time travel and play havoc with longstanding notions of cause and effect. Einstein himself — the author of modern physics, whose theory of relativity established the speed of light as the ultimate limit — said that if you could send a message faster than light, “You could send a telegram to the past.”
Alvaro de Rujula, a theorist at CERN, called the claim “flabbergasting.”
“If it is true, then we truly haven’t understood anything about anything,” he said, adding: “It looks too big to be true. The correct attitude is to ask oneself what went wrong.”
The group that is reporting the results is known as Opera, for Oscillation Project with Emulsion-Tracking Apparatus. Antonio Ereditato, the physicist at the University of Bern who leads the group, agreed with Dr. de Rujula and others who expressed shock. He told the BBC that Opera — after much internal discussion — had decided to put its results out there in order to get them scrutinized.
“My dream would be that another, independent experiment finds the same thing,” Dr. Ereditato told the BBC. “Then I would be relieved.”
Neutrinos are among the weirdest denizens of the weird quantum subatomic world. Once thought to be massless and to travel at the speed of light, they can sail through walls and planets like wind through a screen door. Moreover, they come in three varieties and can morph from one form to another as they travel along, an effect that the Opera experiment was designed to detect by comparing 10-microsecond pulses of protons on one end with pulses of neutrinos at the other. Dr. de Rujula pointed out, however, that it was impossible to identify which protons gave birth to which neutrino, leading to statistical uncertainties.
Dr. Ellis noted that a similar experiment was reported by a collaboration known as Minos in 2007 on neutrinos created at Fermilab in Illinois and beamed through the Earth to the Soudan Mine in Minnesota. That group found, although with less precision, that the neutrino speeds were consistent with the speed of light.
Measurements of neutrinos emitted from a supernova in the Large Magellanic Cloud in 1987, moreover, suggested that their speeds differed from light by less than one part in a billion.
John Learned, a neutrino astronomer at the University of Hawaii, said that if the results of the Opera researchers turned out to be true, it could be the first hint that neutrinos can take a shortcut through space, through extra dimensions. Joe Lykken of Fermilab said, “Special relativity only holds in flat space, so if there is a warped fifth dimension, it is possible that on other slices of it, the speed of light is different.”
But it is too soon for such mind-bending speculation. The Opera results will generate a rush of experiments aimed at confirming or repudiating it, according to Dr. Learned. “This is revolutionary and will require convincing replication,” he said.
Another article, same date:
The trained bicyclists thought they had ridden as fast as they possibly could. But Kevin Thompson, head of sport and exercise science at Northumbrian University in England, wondered if they go could even faster.
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More Personal Best ColumnsSo, in an unusual experiment, he tricked them.
In their laboratory, Dr. Thompson and his assistant Mark Stone had had the cyclists pedal as hard as they could on a stationary bicycle for the equivalent of 4,000 meters, about 2.5 miles. After they had done this on several occasions, the cyclists thought they knew what their limits were.
Then Dr. Thompson asked the cyclists to race against an avatar, a figure of a cyclist on a computer screen in front them. Each rider was shown two avatars. One was himself, moving along a virtual course at the rate he was actually pedaling the stationary bicycle. The other figure was moving at the pace of the cyclist’s own best effort — or so the cyclists were told.
In fact, the second avatar was programmed to ride faster than the cyclist ever had — using 2 percent more power, which translates into a 1 percent increase in speed.
Told to race against what they thought was their own best time, the cyclists ended up matching their avatars on their virtual rides, going significantly faster than they ever had gone before.
While a 2 percent increase in power might seem small, it is enough to make a big difference in a competitive event that lasts four to five minutes, like cycling for 4,000 meters. At the elite level in sports, a 1 percent increase in speed can determine whether an athlete places in a race or comes in somewhere farther back in the pack.
The improved times observed in his experiment, said Dr. Thompson, are “not just day-to-day variability, but a true change in performance.” And they give rise to some perplexing questions.
What limits how fast a person can run or swim or cycle or row? Is it just the body — do fatigued muscles just give out at a certain point? Or is the limit set by a mysterious “central governor” in the brain, as Timothy Noakes, professor of exercise and sports science at the University of Cape Town in South Africa, has called it, that determines pacing and effort and, ultimately, performance?
Until recently, exercise physiologists have mostly focused on the muscles, hearts and lungs of athletes, asking whether fatigue comes because the body has reached its limit.
But athletes themselves have long insisted that mental factors are paramount. Roger Bannister, the first runner to break the four-minute mile, once said: “It is the brain, not the heart or lungs that is the critical organ. It’s the brain.”
Now researchers like Dr. Thompson are designing studies to learn more about the brain’s influence over athletic performance.
For example, Jo Corbett, a senior lecturer in applied exercise physiology at the University of Portsmouth in England, wondered how much competition can affect an athlete’s speed. To find out, he asked cyclists to ride as hard and as fast as they could on a stationary bicycle for the equivalent of 2,000 meters. As he rode, each rider was shown an on-screen figure representing the cyclist riding the course.
Then Dr. Corbett and his colleagues told each athlete that he would be racing against another rider hidden behind a screen. The researchers projected two figures on the screen, one the outline of the rider and the other the outline of the competitor.
In fact, the competitor on the screen was a computer-generated image of the athlete himself in his own best attempt to ride those 2,000 meters.
The cyclists rode furiously through the on-screen race. And, as happened in Dr. Thompson’s experiments, the cyclists beat their best times, finishing with a burst of speed that carried them to virtual victory by a significant length.
Dr. Corbett said the extra effort, above and beyond what the athletes had previously demonstrated, seems to come from the anaerobic energy system, one that is limited by the amount of fuel stored in muscle. The brain appears to conserve the body’s limited fuel to a certain degree, not allowing athletes to work too hard.
But in a race, he said, the brain seems to allow athletes to tap more deeply into energy stores than would ordinarily be permitted. “Competition is able to motivate you to dip further,” Dr. Corbett said.
Related
More Personal Best ColumnsMoney, in contrast, does not increase individual performance, Dr. Corbett said — at least, not in research experiments. Physiologists have asked athletes to go as fast as they can on a course and then offered money if the athletes could beat their own best times. They could not.
Still, there must be a limit to how fast an athlete can go, even with the most intense competition or even with deception. In a new study, Dr. Thompson tried to find what that limit is.
He used the same method as before: Cyclists on stationary bikes raced an on-screen avatar going a bit faster than the cyclist’s own best time. In one group, the only variable was competition. Cyclists were told that the avatar would be going 2 percent faster or 5 percent faster than the cyclist had ever gone.
The other group was deceived. Each cyclist was told to compete against an avatar that would be moving as fast as that athlete had in his best effort. Actually, the avatar was programmed to race 2 percent harder or 5 percent harder. (A 5 percent increase in power translates into a 2 percent increase in speed, Dr. Corbett said.)
The cyclists in the first group gave up from the start when they knew the avatar would be moving faster than they ever had — even when the avatars were going 2 percent harder than the cyclists’ own best times. Instead, the athletes matched their own best efforts.
As had been observed in previous experiments, cyclists in the second group, who were deceived, kept up with their avatars when they were programmed to perform 2 percent harder than each athlete at his best. But 5 percent was just too much: The athletes kept up for about half the race, then gave up.
In the end, their overall pace was no better than it had been in their best effort without the avatar. Some seemed to do even worse than their previous best effort.
“It comes back to the belief system within the athlete,” Dr. Thompson said. Within limits, if an athlete thinks a certain pace is possible, he or she can draw on an energy reserve that the brain usually holds in abeyance.
One lesson, Dr. Thompson said, is that coaches can eke better performances out of athletes by means of small deceptions.
When an athlete has reached a plateau, a coach might tell an athlete in a training session that the course distance is slightly shorter than it actually is, for example, or that his or her speed at each interval is slightly slower than it is.
The new research suggests that this strategy may bring about an increase in performance, and Dr. Thompson said that it has been used to coach elite middle-distance athletes, although he declined to provide details.
But it is a risky approach, he added: Even small deceptions can erode the trust between athlete and coach.
blurb: reduce (or burn an extra) 3,500 calories to lose one pound of body fat — is incorrect and can ultimately doom determined dieters.
`q001. Light moves at 3 * 10^8 meters / second.
The neutrinos beat the light by 60 nanoseconds
How far ahead of the light beam was the neutrino beam?
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What was the percent difference in the speeds of the two beams? Calculate this based on the information provided in this question, and on the distance traveled by the two beams.
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`q002. If one swimmer moves 2% faster than another on a 500-meter race, how far ahead is the faster swimmer at the end of the race?
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If the slower swimmer moves at an average speed of 1.74 m/s during the race, then what is the average speed of the faster swimmer?
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What is the percent difference between the KE per kilogram of body mass, of the two swimmers?
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