1 Bicycles and Stability Bicycles show almost every idea of physics we have seen so far. You push on the pedel which applies torque to the crank sprocket. The chain applies torque to the rear axial gear. The wheels turn and provide stability because of angular momentum conservation. The fork turns the front wheel so you can steer (friction between the tire and road). Brakes apply friction to the wheels to stop your motion. When we were young, it required some practice to not fall over. Not falling over requires stability
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Bicycles and Stability
Bicycles show almost every idea of physics we have seen so far. You push on the pedel which applies torque to the crank sprocket. The chain applies torque to the rear axial gear. The wheels turn and provide stability because of angular
momentum conservation. The fork turns the front wheel so you can steer (friction between the tire and road). Brakes apply friction to the wheels to stop your motion.
When we were young, it required some practice to not fall over. Not falling over requires stability
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Bicycles and StabilityConsider the tricycle. If the tricycle starts to tip over, the center of mass (rider + tricycle) is inside the two wheels This produces a torque brings the tricycle back to level. Notice also that the tip produces an increase in the gravitational
potential energy. With the bicycle, the center of mass quickly falls out of the wheel base and the gravitation potential energy decreases:
Stable Equilibrium: An object is in a stable equilibrium when a small shift increase the total gravitational potential energy
Unstable Equilibrium: An object is in a unstable equilibrium when a small shift decrease the total gravitational potential energy.
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It takes good balance to stay upright on a bicycle stationary bicycle. Why is it easier to stay upright on a moving bike?One thing that stabilizes an moving bike is conservation of angular momentum. When the wheels are turning, the wheels act as 'gyroscopes' to keep the wheel from falling over. However, this is a small effect. The main reason a moving bike is stable is when the bike starts to tip, the upward force of the ground on the front wheel causes a torque on the rotatable front wheel. This 'precession' turns the front wheel away from the direction of the tip which corrects the tip. This Is why you see children learning to ride 'wobble'. It is also why it is relatively easy to ride a bike 'without any hands'.
Bicycles and Stability
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RocketsA rocket or missile is pushed forward by ejecting material out of the rear of the rocket. The reaction force (Newton's 3rd Law) is what pushes the rocket forward.
Consider a very simple rocket consisting of two masses connected together by an initially compressed spring.
If the 'rocket' is initial at rest, then initial momentum is zero.
The speed of the rocket is determined by the exhaust speed.
m2
m1
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Rockets
In a real rocket, the material is normally very hot (and fast) gas which is ejected continuously. We don't need to do themath but the proper expression for a real rocket is:
vr = ve ln ( )
where vr is the final rocket velocity and ve is the exhaust speed. The ln is a 'natural' logarithm (base 'e'). To get the highest speed of the final payload, rockets are often built in stages so empty fuel tanks etc are not carried along to the highest speed. Exhaust speed for chemical rockets are in the range of sevearl km/s. Liquid hydrogen and oxygen has an exhaust speed of 4.5 km/s. Satellite have even used 'ion propulsion which have even hight exhaust speeds.
mrocket + mfuelmrocket
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Newton's Law of Universal Gravity
With one simple mathematical expression, Newton explained how the moon and planets more and how an apple drops. Everything with mass attracts (pulls but never pushes) every other thing in the universe. The force of attraction is proportional to the product of the two masses and the distance between the center of the two objects squared.
F ~
If the mass of either object is doubled, the force is doubles.However, if the distance between the is halved, the force increases by a factor of four. Likewise if the distant between the two object is tripled, the force is one ninth as strong.
m1 m2
d2
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The Universal Gravitational Constant, G
We really want something that gives us the force, not just the proportionality. The constant of proportionality is called the universal gravitational constant, G.
G = 6.67 x 10-11
With the constant we can write Newton's law of gravity as:
F = G
Notice how the units work out with G and the variables in the equations. Because G is so small, gravity is a very weak force. Unless the object is very large (like a planet) the force is very weak.
N m2
kg2
m1 m2
d2
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Satellite MotionIf you throw a ball horizontally, it will travel some distance. Throw it harder and it will go further. If you could throw it at 8 km/s (and ignoring air resistance), it would 'fall' over the horizon and never hit the ground. (What we mean by a horizontal throw is tangent to the earth's surface. Vertical is away from the center of the earth.)
If we give an object more than 8 km/s (orbital speed) then the object will move in an elliptical orbit or higher circular orbit. If the has an altitude of 35,900 km it will take 24 hours to complete on orbit. If the orbit is around the earth's equator, the satellite will appear to be stationary since the earth turns once every 24 hours. This is known as a ’geosynchronous' orbit which is useful for communication satellites.
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Satellite MotionWe can understand elliptical orbits, if we consider energy Conservation. The distance to the center of the earth changes the gravitational potential energy (GPE). The total energy (KE + GPE) is constant. When the satellite is further away from the earth, the kinetic energy (KE) and speed are smaller. When the satellite is near to the earth the GPE is smaller so the speed and KE are greater.
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Escape SpeedIf you send an object directly vertical (not horizontal) it will go up until the KE is converted to GPE, stop and fall back down.If you give it a speed of 11.2 km/s (escape speed) it will never come back down because the KE is larger than the GPE at an infinite distance from the center of the earth. 11.2 km/s is the escape speed from the earth. Only the Apollo astronauts have gone this fast. The escape speed from the Sun is 620 km/s. Only a few man-made objects (Pioneer 10 and Voyager I and II) have gone this fast. Kepler’s Laws of Orbital Motion
Before Newton, Johannes Kepler empirically determined three laws of orbital motion using the data of Tycho Brahe. This laws explained the motion of the planets ('moving stars') which had been a mystery since ancient times.
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Kepler's Laws
Kepler's laws are:
1st Law: Planets move in elliptical orbits around the Sun which is at one focus of the ellipse.
2nd Law: As a planet moves in its orbit, an imaginary line between the planet and the Sun sweeps out equal area in equal time periods.
3rd Law: The period, T, of a planet is proportional to the average distance between the Sun and planet to the 3/2 power. As a formula T = r3/2 ( or T2 = r3 ).
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Kepler's 3rd law is the most useful. It comes from equating the centripetal force on a planet to Newton's law of gravity:
Fgravity = Centripetal force
G =
Since v = 2πr/T
G =
G =
Kepler's Laws
r2
msmp v2mp r
ms
r2
(2πr/T)2
r
ms
r2
4π2rT2
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Kepler's Laws
So we have:
T2 =
T =
This is Kepler's 3rd Law with the constant of proportionality.
4π2r3
Gms
2π r3/2
√Gms
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Special Relativity
We normally think of living in a three dimensional world. To say where something is located, you need three numbers like how far north, east and altitude. In math we would give the x, y and z coordinates. We normally forget about time. But time is just as important as a spatial coordinate. If you look at a distance star, the star you see is how the star looked many year ago. We need to specify time as well as spatial coordinates. When we use this four dimensional description, it is called spacetime. In relativity, we use spacetime.
Inertial Frames
The 'Special' part of special relativity is that we will restrict ourselves to 'frames' or 'view points' moving with constant velocity (constant speed in a straight line) with respect to one another e.g. two cars on a straight highway moving at constant but different speeds.
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Postulates of Special Relativity
The are two postulates of special relativity. This two ideas are all that is required for all the far reaching consequences of special relativity … a beautiful theory.
● All the laws of physics are the same in all inertial frames.
● The speed of light (c = 3.0 x 108 m/s) in free space is the same for all inertial frames. The motion of the source and the observer does not matter.
The first postulate is very basic. The second idea is weird and counter-intuitive to our normal ideas. If you are on an asteroid and Captain Kirk is moving at ½ c in the Enterprise relative to you and a photon torpedo at c is fired forward, you see the photon torpedo moving at c (not 1.5c)
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Time Dilation
One consequence of the speed of light being the same for all observers is that the measurement of time is not the same in all inertial frames. Times gets 'stretched out' when you observe another fast moving frame.
If you measure a time t0 in frame 1, the time that will have
pasted on the clock in frame 2, t, is given by the equation:
t = t0
√1 – (v/c)2
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Time Dilation
The time t0 is called the 'proper time'. It is the time you measureIn your frame.
The factor: is called a γ factor. Most of
What we will do with special relativity is easy if you remember thatMost equations contain this γ factor. For example time dilation is
t = γ t0
Because c is very large (3 x 108 m/s), the γ factor is 1 at anything like the fastest speed we normally encounter. So for time dilationfor normal 'human' speed just says that we observe time thesame for inertial frames move with respect to one another atspeed much smaller than the speed of light.
√1 – (v/c)2
1
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γ Factor
For speeds approaching the speed of light, γ grows quickly. As you can see in the table and graph, at 0.5c γ = 1.15. At 87% of c, γ = 2.0. At the LHC the proton's have a γ of 7461!
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Length Contraction
Space also changes when viewed in another inertial frame moving near c. The length of an object in the direction of motion changes according to:
L = L0 =
This is know as length contraction or Lorentz contraction after Hendrick Lorentz who proposed this contraction to explain the Michelson and Morley experiment. If you see a meter stick moving by you at 87% the speed of light, it will appear to be only ½ meter long to you. In the reference (inertial) frame of meter stick it will still appear to be 1 meter long.
√1 – (v/c)2L0γ
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Momentum and Mass
Momentum is mass times velocity. Nothing can go faster than the speed of light (γ = at c!). So is there a limit to how much momentum something can have? The relativistic moment P is given by:
P =
We normally think of the mass as changing in this equation so the relativistic mass is given by:
m = = m0γ
m0 is called the rest mass. It is the mass of the object in its inertial frame.
√1 – (v/c)2
mv
√1 – (v/c)2
m0
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Equivalence of Mass and Energy
The special theory of relativity also explains where stars get their energy. Probably the most famous equation in the world is:
E = mc2
We can interpret the increase in mass of something moving at the speed of light as the kinetic energy becoming mass.It takes a lot of energy to make a little bit of mass. It works the other way. A tiny amount of mass can become a huge amount of energy. A milligram of something (a few grains of salt) would become 9 x 109 J of energy since c2 is so large (9 x 1016 m2/s2). That is the output of a large power plant for a couple of minutes!
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So all of this sounds pretty weird. Can it really be true? It is commonly observed at particle accelerators with subatomic particles. The experiments and accelerators would not work if relativity was not correct. Remember it only applies to things moving at near light speeds.
At this moment there are muons going through this room.They are created when cosmic rays hit the upper atmosphere a few hundred kilometers up. Muons have a lifetime of about 1 µs (a millionth of a second). So how do they get down here? Answer: time dilation. In their reference frame they live for 1 µs but they are moving at very high speed so to us they love for much longer.
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General Relativity
The special theory of relativity is special since we have limited things to inertial frames. What if the frames of reference are accelerating with respect to one another? In 1915 Einstein published the general theory of relativity. What comes out of general relativity with accelerating reference frames is a new theory of gravity. It reproduces Newton's theory of gravity but applies in extreme cases where Newton's law of gravity breaks down. General relativity only has one postulate know as the equivalence principle:
●Local observation made in an accelerated frame of reference can not be distinguished from observations made in a Newtonian gravitational field.
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General RelativityConsider a 'space ship' floating in space away from any planet. The mass will float around the cabin. If there are no windows (the local observer), and the spaceship accelerates forword, the mass will 'fall' to the floor. Likewise, if the spaceship is on a planet, the mass will fall to the floor. A person inside that can not look outside can not tell the difference between acceleration an a gravitational field.
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General RelativityIf you tossed ball through a hole in the side of the accelerating space ship, what you would see from the outside is a ball moving in a straight line. However, inside the ship, the ball would appear to curve like it was falling in a gravitational field. According to the equivalence principle, inside the ship you could not tell if the path of the ball was bending because the ship was accelerating or because your ship was on a planet with gravity.
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General Relativity
It could be a light beam instead of a ball. Gravity should bend light. After WWI in 1919, an expedition of astronomers measured the bending of light by the Sun during a solar eclipse. Einstein's prediction of the bending was shown to be correct (and different from the Newtonian prediction). Einstein become a 'rock star' when the results made headlines around the world.
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General Relativity
One mystery solved by general relativity was the precession of the perihelion of Mercury. The orbit of Mercury is an ellipse. The major axis of the ellipse slowly rotates. Why had been a mystery for a century. General relativity exactly predicts this rotation. (In the 1800s, a planet inside the orbit of Mercury had been proposed to account for this precision. Of course it was never found but the name was to be Vulcan … Live long and prosper!)
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General RelativityGeneral relativity tells us that mass warps the four dimensional space space-time around the mass. This leads to non-Euclidean geometry. A straight line is no longer the shortest distance between two points like the great circle routes that airplane fly over long distances. In non-Euclidean geometry, light may take a curved path instead of traveling in a straight line. This kind of warped space also explains why the universe is finite but unbounded. Like traveling on a sphere (like the earth) if you head east far enough, you end up coming back to where you were from the west. This warped space time also leads to the idea of black holes in space where nothing (even light) can escape).
