13 April 2010

Physics Problems

Magnetism

Problem 1.
Two long straight wires carrying the same current I and separated by a distance r exert a force F on each other. The current is increased to 4I and the separation is reduced to r/6 . What will be the force between two wires?
Solution:
The force (per unit length) between two parallel wires is
In the present problem the current is the same. Then
The current is increased to and the separation is reduced to then the force is

Optics



Problem 1.
A person stands 40 m from a flag pole. With a protractor at eye level, he finds the angle at the top of the flag pole with the horizontal is 25.0 degrees. How high is the flag pole? The distance from his feet to his eyes is 1.8 m.
Solution:
We draw the picture of the problem, where we show the angle 25 degree, the person's height, and the distance between the person and the pole.
From the right triangle ABC we can find the distance BC (we know the angle and we know the distance AB = 40 m):
The distance OB is equal to the height of the person (which is 1.8 m). Then we can find the whole height of the pole:
Problem 2.
The frequency of yellow light is . Find the wavelength of yellow light. The speed of light is .
Solution:
In this problem we need to use the relation between the frequency , wavelength , and the speed of the light (wave):
Then


Fluids and elasticity

Problem 1.
A cylindrical vessel of radius 0.1 meter is filled with water to a height of 0.5 meter. It has a capillary tube 0.15 meter long and 0.0002 meter radius fixed horizontally at its bottom. Find the time in which the water level will fall to a height of 0.2 meter.
Problem 2.
Calculate the amount of energy needed to break a drop of water of diameter 0.001 meter into 1000000 droplets of equal sizes. The surface tension of water is 0.0072 N/m.
Problem 3.
A force F is applied on a square plate of side L. If the percentage error in determination of L is 3% and that in F is 4%. what is the permissible error in pressure?
Problem 4.
(Physics for scientists and engineers - Serway and Jewett - Chapter 12 - Problem 31)
A walkway suspended across a hotel lobby is supported at numerous points along its edges by a vertical cable above each point and a vertical column underneath. The steel cable is 1.27 cm in diameter and is 5.75 m long before loading. The aluminum column is a hollow cylinder with an inside diameter of 16.14 cm, an outside diameter of 16.24 cm, and unloaded length of 3.25 m. When the walkway exerts a load force of 8500 N on one of the support points, how much does the point move down?

Ideal gas



Problem 1.
How much work has been done by an ideal gas, when the temperature of 5 moles of the gas increases by 2 Kelvin in an isobaric process?
Solution:
The process is isobaric. It means that the pressure is constant. Let us assume that the pressure is equal to and the volume of the ideal gas changes from initial value to the final value . Then by definition, the work done by the gas is
or
But we know that this is an ideal gas. It means that the equation of state of the gas is the Ideal Gas Law. This Ideal Gas Law is valid for both initial and the final states of the gas. Then
where is the number of moles of the gas. Then the work done by the gas is
We know that . Then
Problem 2.
What is an atomic mass of NaOH ?
Solution:
An atomic mass of a molecule NaOH is equal to the sum of
1. atomic mass of Na;
2. atomic mass of O;
3. atomic mass of H.
Atomic mass of Na is 22.99 u ( unified atomic mass unit ).
Atomics mass of O is 16 u.
Atomics mass of H is 1.008 u.
Then the atomic mass of NaOH is

AC current

Problem 1.
A capacitance of is connected to an alternating emf of frequency 100 cycles/second. What is the capacitive reactance?
Solution:
The capacitive reactance is determined by the following expression:
where – frequency, and – capacitance. In the above expression we need to use the correct units for the capacitance: F (farad). Then .
Now we can find the capacitive reactance:


Nuclear Physics

Problem 1.
Given the following masses, find the mass defect and the binding energy of a helium-4 nucleus.
particle mass (kg)
helium-4 nucleus 6.6447X10-27
proton 1.6726X10-27
neutron 1.6749X10-27


Solution:
a. Sum of masses of individual nucleons = 2(1.6726X10-27) + 2(1.6749X10-27)
= 6.6950X10-27 kg
mass defect = 6.6950X10-27 - 6.6447X10-27 = 0.0503X10-27 kg
b. Binding energy = (Δm)c2 = (0.0503X10-27)(3.00X108) = 4.53X10-12 J
(4.53X10-12 J)/(1.60X10-19) = 2.83X107 eV = 28.3 MeV
Problem 2.
a. Given the following masses, find the energy released when U-238 transmutes to Th-234 by alpha decay.
particle mass (amu)
U-238 238.0508
Th-234 234.0436
He-4 4.0026
b. Ignoring relativistic effects, what is the speed of the alpha particle as a result of the alpha decay?
Solution:

a. mass defect: 238.0508 - (234.0436 + 4.0026) = 0.0046 u
energy equivalent: (0.0046 u)(931.5 MeV/u) = 4.3 MeV
b. The values for part a are converted to mks units:
Conversions
quantity symbol conversion factor value
energy released
Ek
1.602x10-19 J/eV
6.88886x10-13 J
mass of Th-234
mTh
1.66053886x10-27 kg/amu
3.8864x10-25 kg
mass of He-4
mHe
1.66053886x10-27 kg/amu
6.6465x10-27 kg
speed of Th-234
vTh
-
?
speed of He-4
vHe
-
?
Conservation of momentum gives us:
mThvTh+ mHevHe = 0
and
vTh+ = -(mHe/mTh)vHe [eqn. 1]
The energy released is in the form of kinetic energy: (1/2)mThvTh2 + (1/2)mHevHe2 = Ek
Substituting eqn. 1 and simplifying:
vHe = √{[2Ek] / [(mHe)(1 + mHe/mTh)]} = 1.4x107 m/s


Problem 3.
What is the wavelength of a 0.217 MeV photon emitted during gamma decay?
Solution:
The energy is converted to Joules: (0.217x106 eV) (1.602x10-19 J/eV) = 3.48x10-14 J

The energy is converted to wavelength:
λ = hc/ΔE = (6.63x10-34 J*s)(3.00x108 m/s)/(3.48x10-14 J) = 5.72x10-12 m


Problem 4.
The radioactivity of milk in one sample was 1750 Bq/L due to iodine-131 with half-life 8.04 days. For comparison, find the activity of milk due to potassium. Assume that one liter of milk contains 1.60 g of potassium, of which 0.0117% is the isotope 40K with half-life 1.28X109 yr.

Solution:

R = [(ln 2)/T][N/A]
where N = Avogadro number [/kmol], A[kg/kmol] is the mass number, ln 2 = 0.693, and T is the half-life in seconds.
One kilogram of a pure radioactive isotope with half-life T[sec] has activity R[Bq/kg]
The unit of activity is Becquerel (1 Bq = 1 decay/sec)
T = (1.28 x 109 yr)( 31,557,600 s/y) = 4.039 x 1016 s
R = [0.693/4.039 x 1016][ 6.0221415×1026 / 40] = 2.583 ×108 Bq/kg
R = [2.583 ×108 Bq/kg][0.00160 * 0.0117/100 kg/L] = 48.35 Bq/L
Relativity
Problem 1.1.


Calculate the velocity of an electron with kinetic energy of 100 keV, 1MeV and 10 MeV.


Problem 1.2.


Calculate the relativistic masses and momenta of the electrons in the three cases in Problem 1.1.


Problem 1.5.


Calculate the velocities at which the electron and proton masses are 2 percent greater than their rest energies.


Problem 1.7.


Two spaceships approach each other, one traveling at 0.3c and the other at 0.6c. What is their relative velocity of approach as seen by an observer on either ship?


Problem 1.9.


Event 1 occurs at x1 = 10m at time t1 = 0. Event 2 occurs at x2 = 600 000 010m at time t2 = 1.8 s. Is there any reference frame where these two events can be reversed so that event 2 occurs before event 1? Prove your answer.


Problem 1.11.


A space shuttle that is 20m long in its rest frame is passing a space docking station which is 110m long in the docking station rest frame. If the space shuttle is moving at 0.6c relative to the station, what will be the length of the docking station to the person on the shuttle? How long will it take the shuttle from the time the front tip of the shuttle reaches one end of the station until the front tip is seen by the person on the shuttle to reach the other end of the station?


Problem 1.13.


An object from outer space moves past the Earth at 0.8c. You measure the length of the object as 3.3m in the Earth’s frame. In the object’s rest frame, what is its length?


Problem 1.15.


A 30-year-old female astronaut makes a trip from the Earth to a star that is 6 light years away at an average velocity of 0.9c. Neglecting the time it takes to turn the spaceship around for the return, how old is the woman when she returns to Earth? How old is her boss, who was 40 when she left? Explain which one aged more than the other and why it was that person who aged more.


Problem 1.17.


Suppose you see that light from a particular type of atom in a distant star that should be at 950nm actually occurs at 525 nm. What is the velocity of the star relative to you, including the direction of the velocity?


Problem 1.19.


Suppose the Concorde flies from Paris to New York, a flying distance of about 3800 miles at a supersonic velocity of 1200 miles/hour. Neglecting the time and distance to reach that velocity, what is the distance as measured by an observer on the plane? What is the time to cover the 3800 miles as measured by a ground observer for a clock on the ground? What is the time as measured by the observer on the ground for the clock on the Concorde?




For solutions to all the problems on this page click here.

Nuclear Physics


1. (a) Use the single-particle shell model to predict the ground-state angular momenta (spin) values and parities for the following nuclei. Explain your reasoning in each case.


7Li, 33S, 44Ca, 69Ga, 80Zr, 123Sb, 175Lu.


(b) For the nuclei listed above, predict the magnetic dipole moments from the shell model.



2. (a) The following nuclei have the spin quantum number quoted, but not the
parity. By comparing the measured dipole moments with the predictions of the
shell model, determine the parity of these states.
J <μ>
43Ca 7/2 -1.3μN
67Zn 5/2 +0.9μN
153Eu 5/2 +1.7μN
139La 7/2 +2.8μN


(b) Compare your findings in 1(b) and 2(a) with the experimental values.
Comment on any differences.


For solutions to the problems click here.

Schedule of Lectures and Problem Classes, Autumn Term 2010
Click here.