12 in 2.54 cm
1 m
15ft ×
×
×
= 4.57 m
1 ft
1 in
100 cm
Conversions:
~+B
~ =C
~
Vectors: A
z
A
A
B
k
C
C
B
x
i
j
y
Cx = Ax + Bx
Cy = Ay + By
Cz = Az + Bz
~·B
~ = |A||
~ B|
~ cos θ = AxBx + Ay By + Az Bz
A
Kinematics:
d~
r
dt
∆~
r
h~v i =
∆t
~v =
d~v
dt
∆~v
h~ai =
∆t
~a =
2
If ~a= const then ~v (t) = ~v0 + ~at and ~
r (t) = ~
r0 + ~v0t + 1
~
a
t
2
1-dim motion (a)
i
0
2
x(t) = x0 + v0t + 1
at
2
v(t) = v0 + at
x
v 2 = v02 + 2a(x − x0)
⇒
(a)
Projectile (b)
~a = −g ĵ: const
v0x = v0 cos θ
vx(t) = v0x
v0y = v0 sin θ
x(t) = x0 + v0xt
vy (t) = v0y − gt
2
y(t) = y0 + v0y t − 1
gt
2
y
y
v
v0
θ
v
a
(b)
a
x
x
(c)
Circular motion (c)
|~v | = constant
v2
|~a| =
;
r
inward
Relative motion
′
~ +~
~
r=R
r
y’
′
~
d~
r = dR
d~
r
dt
dt + dt
y
r’
r
x’
R
′
~
~v = Vframe + ~v
Inertial: Vframe = const or
′
~a = ~a
x
Newton’s laws
1: Inertia (at rest, stays at rest, in motion, remains in motion,
if no force)
~ = m~a
2: F
Σ F =m a
acceleration
~AB
~BA = −F
3: F
resultant of all forces
acting on object
mass
A,B: two different objects
Types of forces
Weight:
~ = −mg ĵ
W
g = 9.8 m2
s
W
~
“Normal”: N
N
W
f
N
W
T
Friction: f~
~ , “opposes” the motion,
f~ k surface, ⊥ N
kinetic or static
P~
F =0
f ≤ µsN , static.
f = µk N , kinetic. Usually, µk < µs
Tension: T~ (pull)
Air resistance:
CρAv 2
D=
2
v=velocity of fall
C= drag coefficient
ρ= air density
A= frontal area
Leads to terminal velocity when D = W .
(W : weight)
1 During a short time interval, the velocity in m/s of
an automobile is given by
v = at2 + bt3
with t in sec. The units of a and b are, respectively
A. m · s2; m · s4
D. m/s3; m/s4
B. s3/m;s4/m
C. m/s2;m/s3
2 Four particles move along the x-axis. Their coordinates
(meters) as functions of time (sec) are
1.
2.
3.
4.
x(t) = 3.5 − 2.7t3
x(t) = 3.5 + 2.7t3
x(t) = 3.5 + 2.7t2
x(t) = 3.5 − 3.4t − 2.7t2
Which have constant acceleration ?
A. all four B. 1 and 2 C. 3 and 4 D. 2 and 3
3 At a stop light, a truck travelling with const. speed 40 ft/s
passes a car as it starts from rest. The car accelerates at
5.4 ft/s2 after this point. How many seconds does it take the
car to catch the truck ?
A. 8
B. 15
C. 24
D. 160
4 An object is thrown vertically upward. While it is rising
A. ~v
B. ~v
C. ~v
D. ~v
and ~a both upward
is up and ~a is downward
and ~a are both downward
is down and ~a is upward
5 A boy on the edge of a cliff 20 m high throws a stone
horizontally outward at 20 m/s. How far away from the
base of the cliff does it hit the ground ? Use g = 10 m/s2
√
A. 10 m B. 40 m C. 50 m D. 50 5 m
6
y z
x
v
N O P Q
M
A cannon fires a projectile as shown. The dashed line shows
the trajectory in the absence of gravity. Points M N O P Q are
separated by 1 sec intervals. Take g = 10 m/s2. The lengths
v : x : y (in m) are
A. 5:20:45
B. 5:10:15
C. 10:40:90
D. 10:20:30
7 A block moves at constant velocity
subject to two forces shown.
An
unknown frictional force is the only other
force acting on the block. It is
3N
5N
A. 0
B. 2N, leftward
than 2N, leftward
C. 2N, rightward
D. slightly more
8 A circus performer of weight W is walking
along a “high wire“ as shown. The tension in
the wire is
A. approx. W
B. approx. W/2
C. much
less than W
D. much more than W
F
M
m
A. mF/(m+M )
9 2 blocks are pushed on frictionless
~ . The magnitude of the force
surface by F
that either block exerts on the other is
B. mF/M
C. mF/(m−M )
D. M F/(M +m)
m
~ /3
A. F
m
B. 0
m
F
~ /2
C. F
10 3 blocks of mass m are sliding
as shown. What is net force applied
to block B ?
~ /3
D. 2F
Work & Energy
2
Kinetic E : K = 1
mv
2
P
Work/energy Kf − Ki = W
~ · d~
~ is CONSTANT
W = F
if
F
Z B
~ · d~
WAB =
F
x generally
A
Work by weight: W = mgd cos θ
2
kd
Work by spring: W = − 1
2
mg
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θ
k
d
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dW ~
Power → P =
= F · ~v for CONST F only
dt
kd
d
Conservation of & Energy
Total energy is always conserved
E = K + U + Eheat + Echem · · ·
Sometimes: conservation of mechanical energy :
E = K + U = CONST (conservative force)
conservative force (e.g. gravity, spring force; counterex.: friction)–
work done between A and B independent of path!
Potential E : Uf − Ui = −W
dU
F =−
dx
Gravitational
Elastic
(by conservative F)
U = −
Z
F dx
U = mgh
1 2
U = kx
2
U(x)
K(x) = E − U (x)
E
x
Momentum
Center of mass: Ftot = M acm
P
Z
Z
mi~
ri
1
1
=
~
rcm =
~
rdm =
~
rdV if ρ = CONST
M
M object
V object
Case of 2 particles: xcm = (m1x1 + m2x2)/M , etc.
Momentum:
particle
(

 P
~ = MV
~cm
system of particles
~
dP
 PF
~
=
ext
dt
p
~ = m~v
P~
p
F = d~
dt
Momentum conservation
If
X
~
F
ext = 0
Impulse
~i = J~ =
∆~
p=p
~f − p
~f = P
~i
P
Z t
f
ti
~ i∆t
F (t)dt = hF
Collisions
1i
2i
1f
2f
1-D, elastic
p1i + p2i = p1f + p2f
K1i + K2i = K1f + K2f
m1 − m2
2m2
v1i +
v2i
m1 + m2
m1 + m2
2mi
m2 − m1
v2f =
v1i +
v2i
m1 + m2
m1 + m2
v1f =
1-D tot. inelastic p1i + p2i = M Vf
v1i +m2 v2i
Vf = m1m
Kf < Ki
1 +m2
y
1
θ1
1
θ2 2
x
2
2-D collision:
p
~1i + p
~2i = p
~1f + p
~2f
pyi = 0 = pyf = m1v1 sin θ1 − m2v2 sin θ2
pxi = m1v1i − m2v2i
pxf = m1v1 cos θ1 + m2v2 cos θ2
Angular motion
s
θ=
r
∆θ = θf − θi
dθ
ω =
dt
dω
α =
dt
If α = CONST
ω = ω0 + αt
2
θ = θ0 + ω0t + 1
αt
2
Angular/Linear
s = θr
vt = ωr
Rolling = Linear + angular
)
s
θ
r
∆θ
hωi =
∆t
∆ω
hαi =
∆t
⇒ ω 2 − ω02 = 2α∆θ
at = αr
Vcm = ω R
v2
= ω 2r
ar =
r
Rotational Kinetic Energy
1 2
K = Iω
2
Z
X
I =
miri2 =
hollow
cylinder
object
I = M R2
1
I = M R2
2
1
I =
mL2
12
2
I = M R2
5
I = Icm + M h2
dm r2
solid
cylinder
holl. cyl.
sol. cyl.
stick
L
stick
sphere
sphere
· · · etc.
axis k cm
Torque
~
~
τ =~
r ×F
τ = r⊥F = rF⊥ = rF sin θ
X
~
τ = Iα
~
Angular momentum

~
l=~
r×p
~
particle
d~l
 P~
τ = dt
(Newton)
Z
W= θf τ dθ
θ1

 L
~ = P ~li
system of particles
~
 P τext = dL
dt
~ = I~
L
ω
~ and ω
⋆ Right hand rule: L
~ are vectors pointing in direction along axis of
rotation given by rh rule: twist fingers of right hand in dir.
of rotation,
thumb points along resultant.
Conservation of angular momentum
P
~f = L
~i
If
~
τ =0
L
P
Precession: In case ~
τ 6= 0, L changes
with time. Know direction of precession
~
from ~
τ = dL/dt.
understand
direction of
precession !
Top view
time ∆ t later
L+ ∆L
∆φ ∆ L
L
Quickie 1
An object moving along the x axis is acted upon by a single
time-varying force. A plot of its velocity is:
Find the sign (+, 0, -) of the work done by the force in intervals
AB, BC, CD, DE, EF.
a) + 0 - + -
b) + + + - -
c) + 0 - - +
d) + + - - +
Quickie 2
A particle of mass m = 10 kg is observed at t = 0 to have a
velocity ~vi = 2î m/s. During the next 10 seconds the particle
is acted on by one force. At t = 10 s the particle is observed
to have a velocity ~vf = 3î + 4ĵ m/s. How much work has been
done by the force?
a) 25 J
b) 105 J
c) 210 J
d) 520 J
Quickie 3
A body at rest (mass m) starts sliding down a plane of length
ℓ inclined at an angle θ from the horizontal. The coefficient of
kinetic friction is µ. What is the object’s speed at the bottom?
a)
c)
q
q
2gℓ(sin θ − µ cos θ)
gℓ(cos θ + µ sin θ)
b)
√
d)
√
gℓ cos θ
2gℓ sin θ
Quickie 4
A sandblasting machine is shooting a jet of sand grains, each of
mass m, speed v0, horizontally at a rock of mass M sitting ona
surface with a coefficient of static friction µs.How many grains
a second must hit the rock to start it moving? (Hint: Collisions
are elastic).
a) M g/mv0
b) M g/2mv0
c) µsM g/mv0
d) µsM g/2mv0
Quickie 7
The rotational kinetic energy of a flywheel is 2.0 × 108 J at a
rotational speed of ω = 460 rad/s. What is the kinetic energy
at a slower speed of ω = 300 rad/s?
a) 8.5 × 107 J
b) 6.0 × 107 J
c) 3.2 × 107 J
d) 2.0 × 108 J
Quickie 8
A dumbell consisting of 2 identical masses m connected by a
massless rod is rotating with angular velocity ω0 about its center
of mass. The distance between the centers of the two masses
is R. A torque τ (out of the screen) is applied, starting at time
t = 0 and lasting for time ∆t. What is the speed of m after that
time?
a) ω0R/2
b) (ω0 + τ ∆t) R
2
c) (ω0 + τ ) R
d)
τ
∆t
ω0 + I R
2
Quickie 9
A spinning skater loses an earring, which flies off in a radial
direction. Which of the following is true?
The skater
A) speeds up rotationally.
B) experiences a force.
C) experiences a torque.
a) A) B) and C) b) only A) and C) c) only A) and B) d) only
B) and C)
Fluids
pressure
Etot
(where z is the coordinate of the system which is oscillating)
(=330 m/s in air at room T)
Decibel scale:
t=0:
Fixed source, moving observer:
Quickie 1
Take g = 10 m/s2 in these problems.
T
x
100
20
75
The diagram shows the masses (in g) and the distances
(in cm) of a mobile. How far from the suspension point
should the 100 g mass be placed?
a) 0.15 cm
b) 7 cm
c) 15 cm
d) 30 cm
Quickie 2
Use G = 7 × 10−11 N m2/kg2 and 4π/3 = 4 in this problem
Two lead spheres of diameter 2 m are just touching each other.
Assume lead (Pb) has a density of 25 g/cm3 or 25,000 kg/m3.
What is the gravitational force between the spheres?
a) 1,000,000 N
b) 1 N
c) 0.2 N
d) 0.000,001 N
Quickie 4
If you are standing on the Moon, and holding a rock, and you
let it go, it will:
a) float away.
c) fall to the ground.
b) float where it is.
d) none of the above.
Quickie 5
The Moon does not fall to Earth because
a) it is being pulled by the Sun and the other planets as well
as by Earth.
b) the net force on it is zero.
c) it is beyond the main pull of the Earth’s gravity.
d) none of the above.
Use g = 10 m/s2 in these problems.
Quickie 1
You observe a simple pendulum clock to have a pendulum length
of 0.25 m and a bob mass of 1.2 kg. What is the period of the
pendulum?
a) 0.3 s
b) 1 s
c) 6 s
d) 12 s
Quickie 2
An object hangs motionless from a spring. When the demonstrator
pulls the mass down a certain distance and holds it there just
prior to releasing it, which of the following statements are true?
a) The sum of gravitational potential energy of the object and
elastic potential energy of the spring has increased.
b) The sum of gravitational potential energy of the object and
elastic potential energy of the spring has decreased.
c) The gravitational potential energy of the object and the
elastic potential energy of the spring are equal to each other.
d) None of the above.
Quickie 4
A particular mass/spring oscillator is observed to have maximum
amplitude A = 5 m and to oscillate at a frequency of f = 7 Hz.
What is the maximum speed that it has during its motion?
a) 5 m/s
b) 35 m/s
c) 70 m/s
d) 220 m/s
Quickie 5
A particular mass/spring oscillator is observed to have maximum
amplitude A = 5 m and to oscillate at a frequency of f = 7 Hz.
What is the maximum acceleration that it has during its motion?
a) 9.8 m/s2
b) 1540 m/s2
c) 9700 m/s2
d) 33,000 m/s2
Quickie 7
You wish to make a simple pendulum clock with a
period of 3 s. How long a string between the pivot
point and the swinging mass should you use?
a) 1.0 m
b) 2.3 m
c) 4.8 m
d) 9.8 m
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θ
mg
Quickie 2
A wave is described by
y(x, t) = 20 cos(6.3x − 22t)
[x,y in m; t in s]
What is the wavelength?
a) 6.3 m
b) 0.16 m
c) 1.0 m
d) 10 m
Quickie 3
A wave is described by
y(x, t) = 20 cos(6.3x − 22t)
[x,y in m; t in s] What is the frequency?
a) 3.5 Hz
b) 22 Hz
c) 0.48 Hz
d) 0.18 Hz
Quickie 4
A wave on a piano string has amplitude 1 mm and a frequency
of 440 Hz. The amplitude is tripled to 3 mm. The frequency is
now:
a) 1320 Hz
b) 147 Hz
c) 220 Hz
d) 440 Hz
Quickie 7
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v1
v2
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m
A weight is hung over a pulley and attached
to a string composed of two sections, each made of the same
material but one having four times the diameter of the other.
String is plucked so that pulse moves along it, at speed v1 in
thick and v2 in thin part. What is v1/v2?
A. 1
B. 2
C. 1/2
D. 1/4