Homework Assignments

i. Do problems with symbols first (introduce them if you have to). Only put in numbers at the end.
ii. Check your answers for dimensional correctness whenever possible.
iii. Examine limiting cases to check your answers (masses to zero or infinity, angles to zero or p/2, etc.)
iv. Homework must show reasoning leading to the final answers in a clear and readable fashion to obtain credit.
v. Unstapled homework will not be accepted!

Assignment 14:

due ***Tuesday***, 5/12

relativity supplements (and 
Chapter R if you haven't yet read it)

problems: chapter R
10 (pion decay, lab frame) (Answers: 49.4 ns, 12.6 m, 6.63 m)
11 (pion decay, pion frame)
33 (energy given velocity)
39 (velocity given energy) (Note: the electron rest energy is 0.511 MeV.)
41 (relativistic vs. non-relativistic kinetic energy)
45 (neutrino mass) Add part (a0) Use problem 39 to show that (since E >> mc^2), mc^2 =~ E Sqrt[2t/T]. Here =~ means approximately equal, T is the light travel time, and t is the time delay for the neutrinos. Using this formula the rest of the problem is simple!

supplemental problems:
S1. If the total energy of a particle of mass m is twice its rest energy (i.e. its kinetic energy is equal to its rest energy), then what is (a) its momentum and (b) its speed? If the momentum of a particle is mc, then what is (c) its energy and (d) its speed? (Answers: Sqrt[3]mc, Sqrt[3]/2 c, Sqrt[2]mc^2, 1/Sqrt[2] c) (Suggestion: It's nice to choose units with c=1 at the beginning, solve the problem, and restore the factors of  c at the end.)

S2. E = mc^2/Sqrt[1 - v^2/c^2] = mc^2 + 1/2 mv^2 + ... Use the Taylor expansion of (1 - x)^(-1/2) to determine the next term.
(Answer: 3/8 m v^4/c^2)

S3. A nucleus of mass m is at rest when it decays with the emission of a photon of energy W and recoils in the opposite direction. Show that the rest mass of the resulting nucleus is m' = m Sqrt[1 - 2W/mc^2] and its recoil velocity is given by v/c = W/(mc^2 - W). (Hint: Use energy and momentum conservation to find the energy and momentum of the decayed nucleus, and use eqns. (R-16) and (R-17).

S4.  A space expedition takes off from the earth and travels at constant speed v=0.9c along a straight path to a star 2 lightyears away. When they arrive they realize they forgot their toothbrushes so they send a radio signal (which travels at the speed of light) back to earth, saying that they are on their way back. They then return immediately along a straight path at speed v=0.5c. (a) Draw a spacetime diagram of this process showing the worldlines of the earth and spaceship and of the radio signal. (b) How much time passes on earth during the round trip? (c) How much time passes on the spaceship during the round trip? (Answers: 6.22 y, 4,43 y)

Pole and barn paradox:  Consider a 20 meter long pole carried by a runner so fast in the direction of its length that it appears to be only 10 meters long inthe rest frame of the earth. This pole can evidently be enclosed in a barn 10 meters long! However, from the point of view of the runner carrying the pole the barn appears to be  contracted to half its length, i.e. 5 meters. How can a 20 meter pole fit in a 5 meter barn? (a) How fast is the runner running? (b) Provide a clear resolution of this paradox in words. (c) Draw a spacetime diagram from the barn's point of view, showing clearly the worldlines of the two ends of the pole, the two ends of the barn, and the simultaneity line corresponding to the barn-moment the pole is enclosed in the barn. (d)  Draw another spacetime diagram, this one from the runner's point of view, showing the same things. Also draw on this diagram the runner's simultaneity lines corresponding to the front of the pole exiting the barn and the back of the pole entering the barn.
Pole and Barn Paradox

S6. Using the k-factor to describe relative motion: The relative motion of two inertial observers can be characterized by their relative velocity,  but a more direct characterization for motion in one dimension is the so-called "k-factor" (also called the "relativistic Doppler shift factor").

If observer O sends out two pulses of light, separated by a time interval T, another observer O' will receive the pulses separated by another time interval T' which is not equal to T. In fact T' must be proportional to T, as can be seen by a simple argument using the homogeneity of spacetime. Hence T' = k T for some constant k, which characterizes the relative motion of O and O'. Refer to the handout for an introduction to the k-factor. For this problem, let's adopt units with c=1, to simplify the writing. You should insert c in the right places in your final formulas using dimensional analysis. 

(a) Draw a spacetime diagram illustrating the definition of k, showing the worldlines of O and O', the two light pulses, and the labeled time intervals.
(b) Show that the relative velocity of O and O' is given by v = (k2 - 1)/(k2 + 1). I think this is simplest to see if you assume that one of the light pulses is sent at the moment the O and O' worldlines intersect, so that light pulse doesn't even appear on the diagram. Also, making O's worldline vertical on the diagram probably makes it easier to see the relations.
(c) Show using (c) that k =  Sqrt[(1 + v)/(1 - v)].
(d) Show that if O1, O2, and O3  are three inertial observers in one dimension, and k12, k23, and k13 are the k-factors for each of the pairs, then  k13  = k12 k23 .
(e) Show using the previous results that v13 = (v 12 + v23)/(1+ v12 v23). This is the relativistic velocity addition formula.
(f) Using dimensional analysis, rewrite the above formulae for v, k, and
v13 restoring the implicit factors of c in the appropriate places.

Assignment 13:

due ***Monday***, 5/5

Chapter  R (Relativity)

problems: chapter 11
18 (Hohmann transfer orbit from Earth to Mars) (Answer: 258 days)
89 (binary star system)
100 (tides)

supplemental problems:
S1. Show that, for a body of mass m in a circular gravitational orbit of radius r about a central mass M >> m,
(a) the kinetic energy K is proportional to 1/r,
(b) the orbital speed is proportional to 1/Sqrt[r]
(c) if the zero of potential energy U is set at r = infinity, then U = - 2K, and E = - K = U/2.
(d) the orbital speed at radius r is related to the escape speed from radius r by v_escape = Sqrt[2] v_orbit.
(e) the angular momentum is proportional to Sqrt[r]

S2. Black hole at the center of the Milky Way: Use the data from the plot of the orbit of the star S2 to estimate the mass of the black hole using Kepler's third law. (You'll need to get the actual size of the orbit from the angular size together with the distance to the galactic center, which is given in the linked article. Note the plane of the orbit is not perpendicular to the line of sight but the article doesn't tell us in which direction. Just read off the semimajor axis as if the projected orbit shown were an ellipse (unless you are really ambitious and want to try to reconstruct the unprojected orbit from the information given and the local of the black hole which should be at a focus of the ellipse!))

S3. Lagrange points: A satellite at one of the Lagrange points will orbit the sun with the same angular speed as the
Earth, staying in a fixed position relative to the Earth. There is currently a NASA satellite at L2, called MAP (microwave anistropy probe), which is taking measurements of the cosmic microwave background radiation. (a) Show that if the angular speed of the satellite's orbit is equal to that of the earth's orbit, then

F_s/(m_s r) = F_e/(m_e D),

where F_s and F_e are the centripetal forces on the satellite and earth respectively and r and D are the distances from the sun to the satellite and to the earth respectively. (b) Write out this equation explicitly using Newton's law of gravity, but instead of writing your answer in terms of r write it in terms of the distance d = D - r from the earth to the Lagrange point. (c) The resulting equation for d is complicated, but one can get a good approximation for the location of L1 and L2 by keeping only the lowest order term in the ratio x = d/D which will be very small on account of the small ratio of the mass of the earth to the mass of the sun. To access this approximation, replace d by xD everywhere in your equation. Then you can show that by dropping all but the lowest order terms in x, the equation implies x = (M_e/3M_s)^1/3, which is very close to 1/100. Show that your answer for d is thus around four times the distance to the moon. (d) Show that you get the same approximation for x for the Lagrange point L2. (Hint: to get the above approximation you will need to use the fact that (1+x)^n is approximately equal to 1 + nx for nx<<1, so (1 + x)^(-3) is approximately 1 - 3x. In another term you will want to approximate (1 + x)^-1 by just 1, since the rest of that term is already small.)

S4. Gravitational Lensing: Since the gravitational force is proportional to the inertial mass, all masses fall the same way in a gravitational field (the "equivalence principle"). Einstein therefore predicted that light would also fall, and therefore starlight passing nearby the limb of the sun would be deflected. His first calculation of the effect was based on Newtonian physics, and gave the result that the deflection angle is 2GM/bc^2, where G is Newton's constant, M is the mass of the sun, b is the radius of the sun,  and c is the speed of light. Later he used general relativity theory for which the result is greater by a factor of two,  4GM/bc^2. In this problem you will derive the Newtonian deflection angle. Consider a particle of mass m and speed v moving in an almost straight line past a mass M, with the distance of closest approach being b. Assuming the deflection is very small, it is given to a good approximation by the angle q = (Dp)/p, where p is the momentum along the direction of motion and Dp is the momentum in the perpendicular direction acquired as a result of the gravitational deflection. Since Dp is to be very small compared to p, we can neglect any equally small change of p (or v) in computing Dp. To compute Dp, observe that it is  equal to the net impulse, which is the integral of F_perp dt over all time. Let's take the particle motion to be in the x-direction, and the perpendicular dispacement b of the line of motion from the center of the mass M to be the y-direction. Then dt = dx/v, and the distance from M to m is Sqrt[b^2 + x^2] when m is at the position (x,b).  (a) Using these ingredients, show that the deflection angle is given by 2GM/bv^2. (Technical hint: Change variables from x to s=x/b to factor out all the b-dependence from the x-integral!) (b) Put v=c in your answer and evaluate the deflection angle for M the mass of the sun and b the radius of the sun. Multipying by 2, show that you get 1.75 seconds of arc, the angle that Eddington claimed to measure in the famous solar eclipse expedition of 1919, thus supporting Einstein's theory.
Assignment 12:

Note: These problems may be extra-tough, so start early and ask questions!

due Friday, 4/25

Chapter 11, Sections 1,2,3

problems: Chapter 10:

47 (spinning coin)  (Answers: In 10^{-5} SI units: 1.33,1.33,1.33,(8.83, -6.17).)
51 (masses, pulley & incline via torque) Add part (d) Solve the problem instead using just energy conservation: express the total mechanical energy in terms of the height y_1 and velocity v_1 = dy_1/dt of m_1,  set this energy equal to a constant, and take the time derivative of the resulting equation. You will find an equation for the acceleration a_1 = dv_1/dt. Compare with part (c). (Another way to solve this would be to get three equations (F=ma for each mass and torque = I alpha for the pulley) for the three unknowns: the acceleration and the two tensions. This
is less efficient if you don't care to know about the tensions.)

52 (spindown of space capsule) (Hint: At what rate is angular momentum injected into the gas? (Nozzle velocity is so high that you can neglect the extra velocity of the gas due to the rotation of the satellite.)) (Answer: 52.4 s)
56 (putty & turntable)  (Answers: w_i / (1+ mR^2/I_0), w_i (explain for credit))
67 (thin bar & blob of putty) (Answers: v_cm  = mv/(m+M), w = mMvd/[(ML^2/12)(m+M) + Mmd^2].)
68 (thin bar & hard sphere) (Answer: L Sqrt[(M-m)/12m])
72 (disk & putty) (Answers: m v_0 b, 2 m v_0 b/(MR^2 + 2m R^2), (m v_0 b)^2/(MR^2 + 2m R^2),1/2 m v_0^2(1 - 2mb^2/(M+2m)R^2))
78 (precessing disk)  (Answers: 3.27 rad/s, 0.164 m/s, 0.535 m/s^2, 24.5 N, 1.34 N)
82 (conical pendulum) (Use mass m, length l, and angle theta, rather than numbers.)  (Answers: In cartesian coordinates (x,y,z) with z the vertical direction, the components fo the angular momentum vector are L =  mvl(cos(theta) cos(wt), cos(theta) sin(wt), sin(theta)), where w = Sqrt[g/(l cos (theta))] and v = l sin(theta) w. The rest is just calculating dL/dt and the torque about the point of support and showing that they are equal.)

supplemental problem:
S1. The earth's axis is tilted at 23.5 degrees away from the normal to the plane of earths orbit. Lets see how this tilt might be the result of a major collision with an asteroid. Treat the original earth as a uniform sphere of mass M and radius R centered at the origin of coordinates and spinning with angular velocity w about the z axis, counterclockwise looking down on the x-y plane from positive z. (a) What was the magnitude and direction of the original earth's angular momentum with respect to the center of the earth? (b) Suppose a huge asteroid of mass m struck the earth at 70 degrees north latitude while moving in the x-z plane (y = 0) with velocity +v in the x direction. What would have been the magnitude and direction of the angular momentum of the asteroid with respect to the center of the earth? (c) Suppose that the asteroid speed v was 100 times as great as the original equatorial speed of the surface of the earth, measured with respect to the center of the spinning earth. Suppose further that after the impact the mass of the asteroid spread out, leading to a spherical earth. What must have been the ratio m/M if the new axis of rotation was tilted at 23.5 degrees to its original direction? (Hint: Just add the two initial angular momentum vectors to find the direction of the final angular momentum vector.)  (Answers: L = 2/5 MR^2 w zhat, 0.940 mvR yhat, 1/540)

Assignment 11:

due Friday, 4/18
See assignment 10 for the one due 4/14 (Monday)

reading:  Finish reading Chapter 10


Chapter 9:

102 (billiard ball translation & spin) (Example 9-14 shows that if h = r + 2/5 r the ball rolls without slipping from the beginning.) (Answer: 5 v_0(h-r)/2r^2.)
132 (pulled spool) We demonstrated this in class. The answer is arccos(r/R). To show this set both the total force and the total torque to zero.

Chapter 10:

1 (T/F cross product questions)
21 (piano stool) (Explain your answer. Add part (b): if the kinetic energy changes, where does the energy go to or come from?
22 (tetherball) (Be careful: The pole has finite radius hence exerts a torque on the ball about the center. See post and puck discussion on page 323.) (Explain your answer to receive credit.)
35 (cross product practice) (Answer: The book's answer to part (c) is wrong. It should be -5 khat.)
39 (cross & dot products)
46 (free particle angular momentum & area)
48 (cancellation of torques)
58 (two disks)
(Answer: 3/5 w_0.)
59 (string through hole)
80 (angular momentum & torque on a particle)
(Answers: 72t J-s khat, 72 N-m khat t.)

Supplemental problem:

A billiard ball initially at rest is given a sharp blow by a cue stick at its midpoint (h=r in the notation of Fig.. 9-63, Prob. 102), and starts moving with velocity v_0. Find (a) the time t when the ball starts to roll without slipping, (b) the distance the ball travels before it begins to roll without slipping, (c) the final speed v of the ball, (d) the fraction of the initial energy dissipated by friction. Assume the ball is uniform and has mass m, radius r, and coefficient of kinetic friction mu_k with the table surface.
(Answers:  (2/7) (v_0/mu_k g), (12/49) (v_0^2/mu_k g), (5/7) v_0, 2/7)

Assignment 10:
due Friday, 4/11 ***extension to 4/14***

reading:  Chapter 10

problems: Chapter 9:

20 (no friction when rolling without slipping)
28 (spin down of turntable) (Answers: 0.134 rad/s^2, 1.75 rad/s, 7.24 rev)
40 (the "sweet spot" of a uniform rod)  (Notes: (i) "short time" means that you should assume the rod rotates very little during the time interval Dt, so that F_0 remains approximately perpendicular to the rod during the interval. (ii) If you've ever played tennis or used a baseball bat you probably noticed that when you hit the ball at just thre right spot of th racquet or bat, you feel no recoil on your hand. This is the "sweet spot". ) (Hints: For part (a), first find the angular acceleration about the pivot using the torque. Note that the analog of impulse for rotational motion is the time integral of torque. For part (b), apply Newton's second law to the tangential acceleration of the center of mass during the time F_0 is applied.)
(Answer: F_0(3x/2L -1))
50 ("The Beast" baton)  (Hint: use the parallel axis theorem for the spheres.)  (Answers: 0.0400 kg-m^2, 0.0415 kg- m^2, ratio 0.964, increase because... )
99 (gears) (Hint: Apply Newton's second law for angular motion to each gear to find two equations. The unknowns are the two accelerations, and the forces the gears exert on each other. However those forces are negatives of each other, and the two accelerations are related via the known radii of the gears, so there are only two unknowns.)
113 (stick falling on ice)
122 (perpendicular axis theorem)
127 (pennies on a falling meterstick)
134 (pulled cylinder) (Note: The mass of the "light drum" can be neglected.)

(T/3)(2r/R -1), (2T/3m)(1 + r/R), r>R/2, same direction as T.

Supplemental problem:

S1. Refer to the racing pendula demo we saw in class. Show that if the extra mass is placed 1/3 of the way from the end the two pendula will fall at the same rate.

Optional problem, not required:
Show that the assumption that kinetic energy is conserved for elastic collisions in three-dimensions in all reference frames implies that momentum is conserved. (Hint: If the velocity of the i^th mass is v_i in one reference frame, it is v'_i = v_i - w in another reference frame moving with velocity w relative to the first. Also, don't forget that for any vector A, the squared magnitude is the dot product of A with itself.)

Assignment 9: (don't forget hw8, which is due Monday 3/31!)
due Friday, 4/4

reading:  Chapter 9

problems: Chapter 8:
63 (handball collision with wall)
73 (block & spring collision) (Hint: At the moment of maximum spring compression the relative velocity of the two masses is zero.)
86 (Lithium-5 decay) Add part (b) What fraction of the speed of light are these speeds? (Answers: 1.74 10^7 m/s, 4.34 10^6 m/s.)  
95 (cue ball and eight ball)
96 (a particular 2d collision) (Answers:  (v_0/2) i + v_0 j, 1/16 m v_0^2)
102 (collision using cm frame)
(Answers:  -3 m/s i {-2 m/s i, 6 m/s i}, {2 m/s i, -6 m/s i}, {-1 m/s i, -9 m/s i}, 42 J)
118 (same impulse, different objects) (Answer:  3/4 v i - v j)
119 (3 to 1 mass ratio and opposite velocities) Solve (b) and (c) of the problem this way: (i) Assume the basketball of mass 3m is moving with velocity v and the baseball of mass m is moving with velocity -v. Find the velocity of the center of mass v_cm. (ii) Find the initial velocities of the basketball and baseball relative to the center of mass. (iii) Using the fact that in the center of mass reference frame the velocities just flip sign, find the final velocities in the reference frame of the floor. Show that the basketball is at rest and the baseball has velocity 2v. Add a part (d) Check directly that this final configuration has the same momentum and kinetic energy as the original configuration.
121 (gravitational sling-shot effect)

140 (falling beads on a balance) (Hint: The average force of the beads on the pan is the negative of the average force of the pan on the beads, which is the average rate of change of momentum of the beads.) (Answer: 31.9 g)

Supplemental problems:
In class we fired an arrow into a hanging box. We previously studied this system in hw6, where you found  the spring constant k for the bow from the force measurements made in class, and computed the initial speed of the arrow using the work-kinetic energy theorem for a given draw distance d of the bow. This week we took another look at the system. Using momentum conservation in the collision we found the relation between the arrow speed v_i and the box speed right after the collision v_f, and using energy conservation we found the relation between the box speed v_f and the height h to which the box rises as it swings up. (See Example 8-14 in the textbook for details.) Finally using geometry we found the relation between h and the horizontal swing distance s = h^2/2L, where L is the suspension length of the box. For this problem you should calculate the initial velocity of the arrow v_i in two different ways: a) from the work-kinetic energy theorem (using k, d, and the masses), and b) from the collision analysis (using s, L, and the masses), and compare. Since not all the work done on the bow is really converted into kinetic energy of the arrow, I suspect method (a) gives a slightly larger result than method (b). Use the numbers: m_a = 30 g, m_b = 3 kg, k = 252 N/m, d = 41 cm, s = 14 cm, L = 121 cm.
Optional problem, not required: (I'll be very impressed if you do it.) (i) Problem 120, and then even more: (ii) For a stack of N balls that strike the floor with speed v, find the masses m_n, n=1,...,N of all the balls so that after the collision all but the top ball will be instantaneously at rest, and find the resulting speed of the top ball. What happens in the limit N goes to infinity? (Answer: m_n/m_1 = 2/(n(n+1)), and v_N/v_1 = N. The answer is simpler than the method I used, so I suspect there is a simpler way to find it. In the limit N to infinity, v_N goes to infinity, and the mass m_N goes to zero. The total mass for N balls is 2(1 - 1/(N+1)) m_1, which converges to the finite value 2m_1. Hence a finite total energy is focused into the kinetic energy of an infinitesimal mass going infinitely fast. The height of the stack is infinite however if the balls all have the same density: the radius of the n^th ball is given by r_n/r_1 = (m_n/m_1)^(1/3) which behaves like n^(-2/3) for large n. The sum of all the r_n thus diverges, since a falloff faster than 1/n is required for convergence of a series of positive terms.)

Assignment 8:
due Friday, 3/21
...but you can also turn it in Monday March 31 if you wish. (There will be a hw9 assignment announced Friday March 21 and due Friday April 4.)

reading:  Whatever you haven't yet read of Chapter 8.

problems: Chapter 8:
6  (True/False momentum questions)
28 (railroad car & grain elevator) (Answers: (a) yes, (b) friction with the falling grain, (c) no...but you must explain!)
39 (center of mass of a semi-circular half-disk). (You can use the hoop result (p. 222) or start from scratch, integrating using polar coordinates.)
52 (3 pieces, initial speed?) (Answer: (c)...
but you must explain!)
53 (exploding shell)
54 (gun, bullet, and block on glider) (Answer to part (c): (m_b/(m_p+m_b)) L.)
55 (object & wedge)
113 (right angle car collision)
127 (rear-end car collision)
Add part (b) : Since horizontal friction acts, horizontal momentum is not conserved (the cars come to rest). Why is it nevertheless OK to use conservation of momentum to analyze this collision?
135 (lifting coiled rope) (The book's answer to part (c) is wrong. The first term should be Mv^2/L.)

Supplemental problem:
S1.  Find the center of mass of an ammonia molecule, NH_3. The three hydrogen atoms form an equilateral triangle, above the center of which sits the nitrogen atom at a distance d. How far from the N atom lies the center of mass, and in what direction? Take m_N = 14 m_H. (Answer: 3/17 d)
Assignment 7:
due Friday, 3/14

reading: Chapter 8

Chapter 6:
67 (U(x) & equilibrium) Add part (b'): Sketch a plot of U(x).
69 (U(x), equilibrium, and speed) Add part (e): Suppose the particle is at x=0 at t=0, and the total mechanical energy of the particle is 2J. Qualitatively (i.e. not quantitatively) describe the subsequent motion over a long period of time, and quantitatively give the maximum positive and negative values of x to which it travels. (You can get the positive one by inspection, but to get the negative one you'll have to use the formula for roots of a cubic polynomial, or inspect a plot or use a numerical rootfinder, or use a spreadsheet or...)

Chapter 7:
23 (playground swing angle)
27 (ball in a vertical circle)
35 (bungee jump)
(Note that the equilibrium length of the bungee cord is 50 m.)
43 (hill climbing) Add part (d): How many "calories" are burned? (1 "calorie" = 1 kcal = 4.184 kJ)
49 (sliding and falling blocks)

66 (lifetime of the sun) Add part (c): How much mass does the sun lose per second? (Use E=mc^2.)
     (Answers to a & b: 3.82 10^26 W, 1.06 10^10 years)
79 (roller coaster)
82 (block on incline with friction and spring) (Answers: 0.989 m, 0.783 m, 1.54 m)

supplemental problem:

S1. The potential energy U(r) corresponding to the gravitational force for a spherically symmetric central mass was derived in class, and is discussed in section 11-3 of the textbook. (a) Reproduce the derivation of U(r) from the force law. (b) Sketch a graph of U(r) with r ranging from the radius R of the sphere to infinity. (c) Suppose a meteor collides with the planet Mars, and ejects a rock of mass m from the surface with speed v.  Using conservation of mechanical energy, show that the minimum speed in order for the rock to escape completely from Mars is v_esc = Sqrt[2GM/R], where R is the radius of Mars.  This is called the escape speed.  Notice that it is independent of the mass m, and independent of the initial direction of the velocity!  (d) Put in the numbers for G, M amd R and determine v_esc and the the corresponding kinetic energy if m is 1kg.  (M = 6.42 10^23 kg, R = 3386 km (this is the average of the equatorial and polar radii)).

Mars meteorites: "Meteoroid impact is the only natural process capable of launching martian rocks to Earth.  To be ejected from Mars a rock must reach the escape velocity of 5.4 km/sec, which is more than five times the muzzle velocity of a hunting rifle.  An impact capable of ejecting the martian meteorites into space would have left a crater of 10-100 km.  The meteorites spent several million years in space before landing at various sites on Earth." For a more comprehensive site see http://www.jpl.nasa.gov/snc/

Assignment 6:
due Friday, 3/7

reading: Chapter 7

Chapter 5:
118 (block sliding off sphere) Add part (0): Using dimensional analysis, first show that the angle at which the block leaves the sphere is independent of its mass, the radius of the sphere, and the acceleration of gravity. (Hint: when the block leaves the sphere the normal force is zero.)
(Answer: arccos(2/3))
Chapter 6:
22 (raising a box) Add part (d) What is the final speed of the box? (Answers: 240J, -177 J,  63.0J, 4.60 m/s)
25 (work from a graph)
34 (two pulleys)  
(Answers: 2h, wh, wh, 2)
39 (force and displacement with vector components)
45 (circular motion another way)
56 (Victoria falls)
(Answer: 879 MW)
57 (box sliding down incline)
72 (lifting the country)
(Answer: 323 km)
80 (work of non-constant force along different paths) (Answers: 0, 78.0 J)
82 (work and power for a non-constant force) (Answers: 9.00 J, -22.1 W)
87 (work of a tangential force)
88 (work of a centripetal force)
(Answers: -0.9 J, 0, 0.463 m/s)

supplemental problem:

S1. In class we loaded a bow string and saw that for 1, 2, 6, 7, 11, 12 kg the corresponding stretches were roughly 3, 7, 24.75, 29, 42,5, 45 cm respectively.  Newton's third law tells us that the force exerted by the bow in these configurations is equal to the weights of these masses. (a) Make a plot of force vs. stretch x for this data using Excel or a graphing calculator or by hand, and fit to a straight line |F| = kx + b.  (In Excel, make a scatter plot and use "Add trendline..." under the chart pulldown menu. Choose linear type and under options check "Display equation".) The data fit a straight line quite well. (The intercept b will be close to zero.) Evaluate the spring constant k. (b) Using your value of k compute the work done in stretching the bow to 40 cm.  (I found 20.1 J.) (c) The mass of the arrow was 30 gm.  Find its launch speed.

Assignment 5:
due Friday, 2/28

reading: Chapter 6

problems: Chapter 5:

14 (racetrack: force on passenger)
16 (loop-the-loop)
19 (terminal speed)
27 (stopping distance)
42 (Lou's bet)
48 (minimum force)
53 (stacked blocks)
71 (two rotating blocks)
73 (swinging child)
91 (max and min speed on banked curve)
109 (upside down carnival ride)
112 (Babylon 5)

Assignment 4:
due Friday, 2/21

reading: Chapter 5

problems: Chapter 4:

blocks & pulley 37 (force vectors & motion)
46 (static equilibrium)
53 (stuck car) 
58 (spring balances)

61 (scale on skateboard)
72 (blocks & massive rope)
81 (housepainter)
83 (mass & pulleys on ledge) (Answer in book is wrong?!)
98 (wedge & block)

Supplemental problems:

S1. What horizontal force F must be constantly applied to M so that m1 and m2 do not move relative to M? Neglect friction. (The table in the figure is supposed to be horizontal.)
S2. In class we fired a pencil part way through a piece of plywood. The pencil had a mass of 4.5 grams, was  travelling at 300 m/s, and came to rest after travelling about 7 cm.  (a) What force, assumed constant, acted on the pencil to bring it to rest? (b) How long did it take for the pencil to stop moving? (c) The pencil was accelerated by a blast of CO2 through a tube. If the tube was 70 cm long what force, assumed constant, acted on the pencil to accelerate it to 300 m/s? (Use your result from part (a). Do not recalculate from scratch!)

S3. An asteroid is discovered to be on a collision course with the earth, and some physics students are desperately trying to devise a way to avoid this calamity. The asteroid has a mass m and is traveling at a speed v toward the collision point, which is at a distance d from the present location of the asteroid. Ignoring gravity, the students determine what constant force F, applied perpendicular to the initial direction of motion during the entire path, is required to deflect the asteroid by one earth radius R by the time it reaches the vicinity of the earth. (a) Using dimensional analysis and the limiting cases of large d or large R, guess their answer up to a numerical constant. (b) What is their result? (Hint: Since the deflection is so tiny, it is very accurate to assume the asteroid travel time is just d/v.) (c) Assuming the asteroid is a sphere of radius 1 km, has a density equal to the average density of the earth, is located at a distance d equal to six times the radius of earth's orbit around the sun, and has a speed v equal to the earth's orbital speed, what is F in Newtons? (You can find the relevant numbers in the textbook on the inside front cover or in Appendix B.)

Assignment 3:
due Friday, 2/14

reading: Chapter 4, Chapter 5-1

Chapter 3:
11 (velocity and acceleration vectors)
27 (relative velocity vector)
29 (path, velocity, acceleration)
48 (vector magnitude and direction practice)
64 (swimming across the current)
72 (centrifuge)
109  (aiming a bullet) (This should be a one or two "dot" problem.) (Note: since the angle of aiming above the horizontal is going to be very small, you can and should use the small angle approximation for the sin function, sin x =~ x.)
110 (vector practice)
114 (hammer off roof)
116 (constant acceleration)

Supplemental problems:
I use boldface for vectors, Ahat for A with a hat over it, D for what is usually the Greek letter Delta, w for what is usally the Greek letter omega. Watch out for similar html accomodations as we go along. (I may be able to insert the Greek fonts later.)

S1. For any vector A we can define a unit vector in the direction of A by Ahat = A/|A|. If A = 3 i + 4 j, what is Ahat?

S2. Consider a particle moving with constant speed in a circle of radius r centered on the origin in the x-y plane.  (a) Carefully draw a diagram showing the circular path of the particle. Use a protractor or a drinking glass or something else so your circle is circular. (b) Use a ruler to carefully draw two position vectors r(t) and r(t+Dt) on your diagram, one at some particular time t and one at t+Dt. Label these and all vectors in this problem.  Assume the particle goes around 1/32 of the circle in the time Dt. (c) Draw the displacement vector Dr between these two times. (d) On the same diagram draw vectors showing the velocity v at t and t+Dt. (e) Draw the change Dv in the velocity vector between these two times. Locate the tail of Dv at the tip of v(t). (f) On the same diagram draw the acceleration vector a(t).  (g) Draw a new diagram showing the circular path, this time including only the three vectors r(t), v(t), and a(t).

S3. Now let's treat the previous problem using calculus. The position vector is r(t) = (r cos wt) i + (r sin wt) j, where w is called the angular velocity. (a) What are the dimensions of w? (b) What is the relation between w and the period T of one cycle around the circle? (c) What is the velocity vector? (d) What is the magnitude of the velocity vector, i.e. the speed v? (e) What is the acceleration vector? Show that it is proportional to the position vector. (f) What is the magnitude of the acceleration vector? Express your answer in two different ways: (i) in terms of r and the angular speed  w, and (ii) in terms of r and the tangential speed v.  (g) Evaluate the position, velocity, and acceleration vectors at the times when wt = 0, Pi/4, Pi/2. Draw a diagram like in the previous problem showing all three vectors at these three times, and compare with part (g) of the previous problem.

Assignment 2:
due Friday, 2/7

reading: Chapter 3
problems: (problems, answers part 1, part 2)
chapter 2:
67 (instantaneous velocity from limit)
76 (landslide)
78 (loose elevator bolt)
88 (flower pot & window)
91 (school zone)
107 (catch up with train)
116 (graphical integration)
118 (differentiation & integration)
122 (non-constant acceleration)

optional challenge: 128 (acceleration as a function of position)

Assignment 1:
due Friday, 1/31

reading: Chapters 1 & 2

problems: (scanned pages: page 1, page 2, answers)
chapter 1:
18 (prefixes)
21 (SI units)
24 (unit conversion)
34 (dimensions)
37 (dimensional analysis)
44 (prefixes & scientific notation)
46 (significant figures & scientific notation)
56 c,d only (follow instructions of prob. 46)