i. Do problems with

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.

v.

**Assignment 14:**

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

reading:

**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!

S1.

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). (

S4.

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 = (k^{2}
- 1)/(k^{2} + 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 O_{1}, O_{2}, and O_{3 } are
three inertial observers in one dimension, and k_{12}, k_{23},
and k_{13} are the k-factors for each of the pairs, then k_{13
}= k_{12} k_{23 .}

(e) Show using the previous results that v_{13} = (v_{ 12}
+ v_{23})/(1+ v_{12} v_{23}). This is the relativistic
** velocity addition formula**.

(f) Using dimensional analysis, rewrite the above formulae for v, k, and v

**Assignment 13:**

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

reading:

**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.)

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

reading:

**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 d**L**/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 **z**hat, 0.940 mvR **y**hat, 1/540)

**Assignment 11:**

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

**problems: **

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 **k**hat.)

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) (*Answer*s: 72t J-s **k**hat, 72 N-m **k**hat t.)

S1. 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.

(

**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.)

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

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.

------------------------------------------------------------------------------------------------------------

Show that the assumption that kinetic energy is conserved for elastic collisions in three-dimensions in all reference frames implies that momentum is conserved. (

**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)

S1. 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.

------------------------------------------------------------------------------------------------------------

**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.)

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. (

due Friday, 3/14

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...)

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.)

(

79 (roller coaster)

82 (block on incline with friction and spring) (

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/

due Friday, 3/7

**reading: **Chapter 7

**problems:**

**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.

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)

due Friday, 2/21

**reading: **Chapter 5

**problems:** Chapter 4:

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

**problems:**

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 (

116 (

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 D**r** between these two times. (d)
On the same diagram draw vectors showing the velocity** v** at
t and t+Dt. (e) Draw the change D**v** in the velocity vector between
these two times. Locate the tail of D**v** 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.

**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)

**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)