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Physics 11 – Special Relativity, Nuclear Physics and Radioactivity
Special Relativity:
A. Events and Inertial Reference Frames
· Reference Frame
Each observer is at rest relative to his own reference frame.
· Inertial Reference Frame
An inertial reference frame is one in which Newton’s law of inertia is valid. That is, if the net force acting on a body is zero, the body either remains at rest or moves at a constant velocity. In other words, the acceleration of such a body is zero when measured in an inertial reference frame.
B. The Postulates of Special Relativity
· The Relativity Postulate
The laws of physics are the same in every inertial reference frame.
· The speed of Light Postulate
The speed of light in a vacuum, measured in any inertial reference frame, always has the same value of c, no matter how fast the source of light and the observer are moving relative to each other.
According to the relativity postulate, any inertial reference frame is as good as any other for expressing the laws of physics.
v < c / v » cTime / Dti /
Length / Li /
Momentum / /
Kinetic Energy / /
Addition of Velocity / /
C. Time Dilation (moving clocks run slowly)
Dt : time interval measured by an observer who is in motion with respect to the events and who views them as occurring at difference places
Dti : proper time interval between two events, as measured by an observer who is at rest with respect to the events and who views them as occurring at the same place
v : relative speed between the two observers
c : speed of light in a vacuum
e.g. The spacecraft is moving past the earth at a constant speed that is 0.92 times the speed of light. The astronaut measures the time interval between successive “ticks” of the spacecraft clock to be 1.0 s. What is the time interval that an earth observer measures between “ticks” of the astronaut’s clock? (2.6 s)
e.g. Alpha Centauri, a nearby start in our galaxy is 4.5 light-years away. If a rocket leaves for Alpha Centauri at a speed of 0.95 c relative to the earth, by how much will the passengers have aged, according to their own clock, when they reach their destination? Assume that the earth and Alpha Centauri are stationary with respect to one another. (1.4 years)
e.g. The average lifetime of a muon at rest is 2.2 ´ 10-6 s. A muon created in the upper atmosphere, thousands of meters above sea level, travels toward the earth at a speed of 0.998 c. Find, on the average, (a) how long a muon lives according to an observer on earth, and (b) how far the muon travels before disintegrating. ((a) 35 ´ 10-6 s, (b) 1.0 ´ 104 m)
Practice:
1. (a) What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at 0.60 c with respect to the laboratory? Its mean lift at rest is 2.2 ´ 10-6 s. (b) How far does a muon travel in the laboratory on average, before decaying? ((a) 2.8 ´ 10-6 s, (b) 500 m)
2. An astronaut whose pulse remains constant at 72 beats/min is sent on a voyage in a spaceship, and his pulse is measured by a stationary observer on Earth. What would his pulse beat be when the shop is moving, relative to the Earth, at (a) 0.10 c and (b) 0.90 c? ((a) 0.014 min (72 beats/min), (b) 0.032 min (31 beats/min))
D. Length Contraction (moving objects are shorter (in the direction of motion))
L: the relativistic distance as measured in the moving spaceship
Li : the proper length; it is the length (or distance) between two points as measured by an observer at rest with respect to them.
v : relative speed between the two observers
c : speed of light in a vacuum
e.g. An astronaut, using a meter stick that is at rest relative to a cylindrical spacecraft, measures the length and diameter of the spacecraft to be 82 and 21 m, respectively. The spacecraft moves with a constant speed of 0.95 c relative to the earth. What are the dimensions of the spacecraft, as measured by an observer on earth? (26 and 21 m)
e.g. A rectangular painting measures 1.00 m tall and 1.50 m wide. It is hung on the side wall of a spaceship which is moving past the Earth at a speed of 0.90 c. (a) What are the dimensions of the picture according to the captain of the spaceship? (b) What are the dimensions as seen by an observer on the Earth? ((a) 1.00 m by 1.50 m, (b) 1.00 m by 0.65 m)
E. Mass Increase as well as Momentum
m : the relativistic, or moving, mass, which will be measured to have in a reference frame in which it moves at speed v.
mi : the rest mass of the object: the mass it has as measured in a reference frame in which it is at rest.
v : relative speed between the two observers
c : speed of light in a vacuum
e.g. Calculate the mass of an electron when it has a speed of (a) 4.00 ´ 107 m/s in the CRT of a television set, and (b) 0.98 c in an accelerator used for cancer therapy. The rest mass of an electron is 9.11 ´ 10-31 kg. ((a) 9.19 ´ 10-31 kg, (b) 4.58 ´ 10-31 kg)
When the speed approaches the speed of light, an analysis of the collision shows that the total linear momentum is not conserved in all inertial reference frames if on defines linear momentum simply as the product of mass and velocity. In order to preserve the conservation of linear momentum, it is necessary to modify the classical definition of momentum . The theory of special relativity reveals that the magnitude of the relativistic momentum must be defined as
e.g. The particle accelerator at Stanford University is three kilometers long and accelerates electrons to a speed of 0.9999999997 c, which is very nearly equal to the speed of light. Find the magnitude of the relativistic momentum of an electron that emerges from the accelerator, and compare it with the non-relativistic value. (1 ´ 10-17 kg×m/s, 4 ´ 104)
F. The Equivalence of Mass and Energy
One of the most astonishing results of special relativity is that mass and energy are equivalent, in the sense that a gain or loss of mass can be regarded equally well as a gain or loss of energy.
· Total Energy
· Rest Energy
· Kinetic Energy
e.g. An electron (mi = 9.109 ´ 10-31 kg) is accelerated from rest to a speed of 0.9995 c in a particle accelerator. Determine the electron’s (a) rest energy, (b) total energy, and (c) kinetic energy. (8.19 ´ 10-14 J, 2.59 ´ 10-12 J, 2.51 ´ 10-12 J)
e.g. A meson (mi = 2.4 ´ 10-28 kg) travels at a speed 0.80 c. What is its kinetic energy? (1.4 ´ 10-11 J)
G. Relativistic Addition of Velocities
e.g. Imagine a hypothetical situation in which the truck is moving relative to the ground with a velocity of 0.8 c. A person riding on the truck throws a baseball at a velocity relative to the truck of 0.5 c. What is the velocity of the baseball relative to a person standing on the ground? (0.93 c)
The Nucleus and Nuclear Applications:
q Description of the Nucleus
- The atom is neutral. The proton is positively charged with one unit of elementary charge. Its mass is approximately one atomic mass unit, u.
- Number of protons is called the atom’s atomic number, Z.
- Number of protons = Number of electrons (in a neutral atom)
- A neutron is a particle with no charge and with a mass almost equal to that of the proton.
- The sum of the numbers of protons and neutron is equal to the mass number, A.
q Isotopes
- Atoms with the same atomic number but different mass numbers are called isotopes; the same number of protons, but different number of neutrons.
- All isotopes of an element have the same number of electrons around that nucleus and be have the same chemically.
e.g. and
Practice:
1. An isotope of oxygen has a mass number of 15. The atomic number of oxygen is 8. How many neutrons are in the nuclei of this isotope? (7)
2. Three isotopes of uranium have mass numbers of 234, 235, and 238 respectively. The atomic number of uranium is 92. How many neutrons are in the nuclei of each of these isotopes? (142, 143, 146)
3. How many neutrons are in an atom of the mercury isotope ? (120)
q Radioactive Decay
In 1899 Rutherford discovered that uranium compounds produce three different kinds of radiation. He separated the radiations according to their penetrating ability and named them a (alpha), b (beta), and ¡ (gamma) radiation.
- a Decay
The emission of an a particle is a process called a decay. The a radiation can be stopped by a thick sheet of paper and a particle is the nucleus of a helium atom, . Since a particles contain protons and neutrons, they must come from the nucleus of an atom. The nucleus that results from a decay will have a mass and charge different from those of the original nucleus. The mass number of an a particle is 4, so the mass number of the decaying nucleus is reduced by 4. The atomic number of a particle is 2; therefore, the atomic number of the nucleus is reduced by 2.
- b Decay
b particles are negative electrons emitted by the nucleus. b particles can be identified as high-speed electrons. 6 mm of aluminum are needed to stop most b particles. Since the mass of an electron is a tiny fraction of an atomic mass unit, the atomic mass of a nucleus that undergoes b decay is changed only a tiny amount. The mass number is unchanged. The nucleus is changed to a proton within the nucleus. An unseen neutrino accompanies each b decay. In the other words, in b decay, the atomic number is increased by one while the mass number is not changed.
- ¡ Decay
¡ radiation results from the redistribution of the charge within the nucleus. The ¡ ray is a high energy photon and several centimetres of lead are required to stop ¡ rays. Neither the mass number nor the atomic number is changed when a nucleus emits a ¡ ray in ¡ decay.
a particles and ¡ rays are emitted with a specific energy that depends on the radioactive isotope. a particles, however, are emitted with a wide range of energies. Radioactive elements often go through a series of successive decays, or transmutations, until they form a stable nucleus. For example, undergoes fourteen separate transmutations before the stable lead isotope is produced.
Practice:
1. Write the nuclear equation for the transmutation of a radioactive radium isotope, , into a radon isotope, , by the emission of an a particle.
2. Write the nuclear equation for the transmutation of a radioactive lead isotope, , into a bismuth isotope, , by the emission of a b particle and an antineutrino.
3. Write the nuclear equation for the transmutation of a radioactive radium isotope, , into a radioactive radium isotope by the emission of an a particle.
4. Complete the following nuclear equations.
(a)
(b)
q Half-Life
The half-life of a radioactive material is defined as the time it takes for half the original amount of the radioactive material in a given sample to decay.
e.g. The half-life of is 5730 years. If at some time a piece of petrified wood contains 1.00 × 1022 nuclei of , then 5730 years later it will contain only 0.500 × 1022 of these nuclei. After another 5730 years it will contain 0.250 × 1022 nuclei, and so on.
e.g. If 100 g of a radioactive material with half-life of 7 years are stored in a cave for 2401 years, how many grams will be left at that time? (12.5 g)
q Nuclear Reaction
A nuclear reaction is said to occur when a given nucleus is struck by another nucleus, or by a simpler particle such as a γ ray or neutron, so that an interaction takes place.
Ø Nuclear Fission
In nuclear fission, the total mass of the products is always less than the original mass of the reactants. Nuclear fission occurs when a heavy nucleus splits, or fissions, into two smaller nuclei. The lost mass is transformed into energy, electromagnetic radiation, and the kinetic energy of daughter particles.
Nuclear fission occurs inside sun’s core, earth’s core, and hydrogen bomb.
- Chain Reaction
One neutron initially causes one fission of a uranium nucleus; the two or three neutrons released can go on the cause additional fission, so that process multiples. Therefore, a chain reaction may also refer to neutrons producing more neutrons.
- Nuclear Bomb
If the chain reaction is not controlled, it will proceed to rapidly and possibly result in the sudden release of an enormous amount of energy (an explosion). An uncontrolled fission reaction is the principle behind the nuclear bomb.
- Nuclear Reactor
A nuclear reactor is a system designed to maintain a self-sustaining chain reaction. Nuclear reactors using controlled fission events are currently being used to generate electric power.
The most significant difference between a nuclear reactor and a nuclear bomb is the rate at which mass is converted into energy.