Unit 2 Forces Review
SEP4 – Graphical Review Problems
1) Use the grid to draw an accurate FBD for the following situations:
a) You pull a piece of tire across the floor at a constant 2 m/s.
b) You pull the same piece of tire across the floor at a constant 4 m/s.
c) You pull the same piece of tire across the floor so it accelerates at a rate of 2 m/s .
d) You pull the same piece of tire across the floor so it accelerates at a rate of 4 m/s .
2
2
a
qtr
a
Fn
more Fp
FETE
a
q
FETE
meansmore
FYI
Y
a
a
qFn
FTII Fp
tis
2) A student pulls a frictionless hoverpuck using
a spring scale. The student graphs the data
below. How should you use the graph to
accurately determine the mass of the puck?
SLOPE X
qtr
t
same
Y INT
SLOPE Free to
FI Mt Fnet FggÉ
Taketheinverse oftheslope
Fret 2Fut
3) The student does a new experiment where he pulls a hoverpuck
with a constant force but sees how changing the mass affects
the acceleration. What would you expect this graph to look
like?
a) Label the axis with units and
b) Sketch the graph to the right.
and
mckg
4) Two masses (A and B) are attached via a string. The mass of A is less that the mass of B. Use the
grid below to draw and accurate FBD for objects A and B if the surface…
a) is frictionless.
b) has friction and the system stays at rest.
c) has friction and the system starts from rest
and then speeds up.
Fn
Fun
I
Ft
Fg
F
Ft
Fga
Ft
Ft
Ft
Fg
Fga
Ft would
Fun
be more
Ft
Fg
Ft than in
Fga
5) A block rests on a rough horizontal surface. A
student attaches a force sensor to the block and
then pulls to the right. The student makes the
above graph.
gtfo
a) Label what the motion is and the type of
friction is present for each section.
Moving
Kinetic Friction
b) Look up the formula for friction on the equation sheet. Write it below. How does the equation
relate to the graph above?
FE MEN
EffMFy
Once the Ff becomes
kinetic then it stays constant
Maximum
In graph Ffs
Friction
holdsuntilthe static
possible
MaxFesisreached
6) A brick and piece of wood are pulled horizontally at a constant 1 m/s across a rough surface and
pull force is measured. Mass is then added to each object and the experiment is repeated.
a) Draw an accurate FBD for the brick and
wood below for one of the data points
(be sure and label which data point you
used).
Fr
a
p
I
Fp
Fg
b) Using the graph, what would be the most accurate way to determine the coefficient of kinetic
friction?
Y
Ff
SLOPE X
t
Y INT
SLOPE Fn to
Fr M Fn
7) A block slides up a rough incline. Draw an FBD for this situation.
Clearly indicate any forces that are the same magnitude.
Fr
Fgy
8) A shopping cart is pushed for a long
time. The size and direction of the
force can change over time. For this
problem let’s call to the right positive
and to the left negative.
Fox
Draw an FBD for the following times.
For each FBD make sure it is clear the
direction of the net force.
a
d
In
a
f
F
Fn
E Fp
FI
Fg
F
Fg
In
LessFreethan
Fp
Fg
if
e
a) 5 seconds:
b) 12 seconds:
c) 20 seconds:
d) 30 seconds:
e) 35 seconds:
f) 50 seconds:
b
in
and
Fg
do
c die
In
F E Fp
Fg
Samoa
same Fnet
sameFBI
9) A heavy crate is pushed across the floor and accelerates to the left.
a) Draw two FBDs for the crate: one with the angle θ, and one with
an angle 2θ.
Fox
FM
É
F
I
Fa
F
Ff
Fry
F
Fp
Fg
b) For each force, state whether it increases, decreases, or remains the same if the angle were
to change from θ to 2θ.
Mass is thesameso
MoreO More Fpy More Fn
More
LessFpx
Ff
Fg isthesame
c) How does the net force on the crate when the angle is θ compare to the net force on the
crate when the angle is 2θ? Explain.
Fnet Fpr Ff
Sincethere is less Fpr and more Fr theFnet is less
10)A person has a choice of either pushing or pulling a sled at a constant velocity, as the drawing
illustrates. Friction is present. If the angle is the same in both cases, does it require less force to
push or pull the sled? Explain.
Pullin
Pulling up reduces Fn Ft
SEP5 – Mathematical Problems
11)Some blocks that are hooked together are pulled to the left across a frictionless surface.
15050.15kg
a)
What is the total mass of the system, in kilograms? ____________________
b)
Draw an FBD for each block as well as the system of both blocks.
c)
É
i Éi
If the force on block B by block A is to the right and has a magnitude of FB, what is the pull
force by the hand, in terms of FB? Show your work. Hint: use proportional reasoning.
Thismuchforce
aE
is
50 FB Fp
y they
o
takes lathe
12)A mass A is hung by a string with is connected to the cart B.
Assume no friction between the cart and the track.
a) Determine the acceleration of the cart if 3mA = 5mB. Give
your answer in terms of g, the freefall acceleration.
Ma MB
Liga
13)A student has a weight of 578 N. While accelerating up in an elevator, the scale reads 623 N.
a) Calculate the mass of the student.
578N 9845459.01
Fg Mg
b) What is the normal force exerted on the
student while riding in the elevator?
623N
c) Calculate the acceleration of the student while in the elevator.
Fet F F m a
623N 578N69kg a
19 276
14)The engine of a 1,250 kg car provides a force of +4,500 N.
a) If the car accelerates at a rate of +3 m/s2, what is the total resistive force (wind and friction)
acting on the car?
FA Ff ma
4500
4500 Ff 1250713
Fa
TFM
Ff 7S
b) How much engine force does the driver need to apply if they wanted to travel at a constant
Fnetcx
velocity instead?
Fnetoforconstantvelocity
4500 750 3750
constantvet
EngineneedstodecreaseFaby3750Ninordertomorew
3750N FAWillalsobeSONmakingFneta FAFf ON
Ifittby
15)An object slides down 25° incline at a constant speed. Determine the force of kinetic friction.
Show all steps.
I
ÉÉÉÉ
Fn Fg mycost Mamycost mosint
Marcos sine
Fr Fgx
mermgs.int
Mkmgcost mgsi.no
Me
16)5 N of force is required to keep a block moving at a constant 1 m/s. How much force must be
applied to the block to keep it moving at 2 m/s?
I Ff
same
of
is
independent
V
17)A 1 kg block starts from rest and is pushed with a force of FA. The coefficient of friction
between the block and the ground is µ. All forces are applied for a time t. Derive an equation
for the speed of the block at time t in terms of FA, µ, g, and t. Show all work.
qtr
ftp.y
tf
Fn FsFnettmaxV
attVoFn
V Fa ng t
my Fa Fa Ma
Fr a g Fa mFn Gkga
Fn g
Fa ng a
18)
É
fFgs
Show your work.
FfEMFNzFhet
FEMMag
MsysAFgiIFs
Fgsm.a
Mig MMag Mag Cmitmatma
19)
Initiate
L
t
Fnet
A
Fnet Mia
Ma
Mag Mig Mig ChitmatMDa
9 79
1.452 45 easy ay
Fnety Muay
Mug Femuay
4dg F e41.4
39.2 Fa 5.6
FI 33.6N
Mig Fema
169.8
Fi
1.4
9.8 17 1.4
8.4N
FE
Sign error. Answer is 11.2 N
If accel is positive, then Ft
is positive and Fg is –.
20)A 50 kg person stands on a bathroom scale in an elevator (800 kg). If the scale reads 600 N,
determine the elevator’s acceleration. Is the person accelerating up or down? Is the person
moving up or down?
600N
IF
Fet ma
Can't tell direction
110 509
of V from a
600490 50a
OFF 50.9.8 490N
a
zzz ay
21)A block is moving 10 m/s slides over a rough surface with a µk. The block stops in a distance of
d. If the surface is changed so the µk is half as big, what would the new stopping distance be.
Ft
ME
tis
Fnet Ma
Ff
MA
K O
X d
V
022
Vodat
V o
a ng
M Fn ma
Mmg ma
ng a
flat
Vitzast
EDGE d
d
2d
22)A small car and an SUV are at a stoplight. The car has a mass equal to half that of the SUV, and
the SUV's engine can produce a maximum force equal to twice that of the car. When the light
turns green, both drivers floor it at the same time. Which vehicle is has the faster acceleration?
Prove your answer using equations.
IE a
Etana
atthesamerateduetothe
I
EE
massdoubles accelwillbesame Fneticar
bloits a ratioof FA M
ma
M
FA A
A
f
Fetisov ma
LFA 2m A
HIM
FAM
a
FAM
23)Two blocks, A and B, are connected by a massless string over a massless pulley. What is the
magnitude of the acceleration of block B if mA = 4mB. Give your answer in terms of g the
acceleration of an object is free fall.
FnetsystMsysa
go
If
q
Fa Fy Fatma a
E
Mag Mag
matmja
4mg may 4m.tm.la
37,9 5NBA 3g a
I
24)A student wants to slide his iPad across the lab table to another student. The coefficient of
kinetic friction, μk, for the iPad case and lab table is 0.55 and the coefficient of static friction, μs,
is 0.6. The iPad has a mass of 0.662 kg and the case 0.0127 kg.
a) What is the minimum amount of force needed to make the iPad slide?
I.gg
FpFFs MsFn MsFg Ms Miraatmase g
0.660.662 0.1279.8
0.6 0.789 9.8 4.64N
b) If the student is using 5 N of horizontal force to slide the iPad, what is its acceleration?
5 0.55 0.78979 8
Fnet IMA
Fp Ee
ma
0.789a
9 0.94712
4.64N
Nd
c) How long will it take the iPad to get across the lab table, which is 1 m wide, if the student
continues to push it with this 5 N of force?
X tat't rot Xo
1 1260.947 E
9mm
YÉÉÉ
ym
t 1.45s
25)A box is pulled to the right with an acceleration of 1.5 m/s2 and a force of 10 N. The box has a
mass of 5 kg.
a) What is the coefficient of kinetic friction?
Fnet Ma
Ff 2.5N M 0.51
Fp Fema
5.5
50.5711
5
5 17
b) How fast will the block be moving after 3.5 s if it was already moving to the left with a speed
of 3 m/s before the force?
I
V at TV
V 1.563.57 3
ÉÉÉ V 8.255
SEP6 – Verbal Problems
26)An object is moving to the right, and experiencing a net force that is directed to the right. The
magnitude of the net force is decreasing with time. Describe and explain the motion of the
object.
• Fnet causes ACCELERATION
• If Fnet is pointing to the right and decreasing, then acceleration is pointing to
the right and decreasing.
• Since velocity points right and acceleration points right, the object is still
getting faster and faster, just less rapidly.
27)A person celebrates all 3’s on their physics test by going skydiving!
a) What is the 3rd Law pair of the force of gravity (Fg) acting on them?
The force of gravity is on you by the Earth. The 3rd law pair would be a force of gravity on the Earth
by you with exactly the same force. (The Earth has WAY more mass/inertia than you so it does not
really accelerate at all when you are in free fall, while you accelerate at a rate of 9.8 m/s2).
Remember, 3rd law is NOT “equal and opposite reactions,” it is “equal and opposite forces.”
b) If these 3rd Law force pairs of forces are equal in size and point opposite direction, why
doesn’t the net force of the person equal 0 N?
3rd Law force pairs do not balance each other out because they act on different
objects (forces only balance if they act on the same object).
c) When you are in free fall, why do you get pulled down to the ground instead of the ground
getting pulled up to you?
Though the force of gravity on both you and the Earth might be the same, the
MASS is so much larger for the Earth that the force of gravity can not cause as
much acceleration as it can for you.
28)A car driving along hits a bug.
a) How do the forces the car and bug exert on each other compare?
b) How do their accelerations as a result of these forces compare? Why?
The force on the car by the bug is equal in size and opposite in direction to the
force on the bug by the car. But, the bug has MUCH less mass/inertia so it
accelerates at a much higher rate. This acceleration is so high that the bug
goes splat. The acceleration of the car is nearly zero because it has so much
mass/inertia
29)You pull up at an angle on a box and pull it across a rough floor. Would the normal force be
equal to, more than, or less than the weight of the box? Why?
• The upwards component of pull would decrease how hard the surface has to push up to
support the box.
• Therefore the normal force would be LESS than the weight of the box.
30)Newton’s 1st Law tells us that a net force is not required for an objects to stay in motion.
However, anyone who has pushed a shopping cart would tell you that their push force IS
required for the cart to keep moving, What are the people confusing about what Newton’s 1st
Law states and what they are experiencing?
No force is necessarily required. An object moving at a constant velocity requires no
net force to maintain that velocity. Why so many people get this wrong is that often in
real life friction will act on moving objects. This means that in order for the net force to
be zero you must have some force acting on the object to cancel out friction
31)Explain what happens to the motion of an object moving to the right (with no friction) when a
net force points…
a) To the right?
The net force points in the same direction as the acceleration. If the acceleration
and velocity point the same direction the object will go faster.
b) To the left?
If the acceleration and velocity point opposite ways the object will get slower.
32) (SEP4,5,6) A box of m is moving at a constant velocity of v to the right while being pushed. A
bucket with mass of 2m is moving at the SAME constant velocity of v to the right while being
pushed. Assume the bucket and the box are made of the same material and slide on the same
surface. Is the box or the bucket being pushed with more force?
Picture/FBD
g
Fn
Égypas
I Fg
Select Evidence (Given This…)
Eiji
to maintainconstant velocity is
pp
Claim
Equations and/or physics concepts:
Fr F
i
Fir my Frush F O
Fpush MFn
Fpush Mmg
Reasoning (Means that…)
then a bucket with Twice
the mass will require
TWICE the push force
The Bucket requires more force
U2 Forces - Practice Test
Name: ______________________________________
1. (SEP4 – Graphs) The block slides down the ramp at a constant
speed. Draw an FBD, making sure to indicate any forces or
components of forces which are equal in magnitude.
Q
Fna O
rFN
Fyffe
2. (SEP4 - Graphs) The hanging mass (mh) is smaller than
the mass on the table (mt). There is an additional force
of tension, FTB, acting on the mass on the table, causing
the system to accelerate in the direction of FTB. Friction
is NOT negligible. Draw an FBD for the hanging mass
and draw an FBD for the mass on the table. Mark any
forces that are equivalent with tick marks and be VERY
obvious about relative sizes of forces.
ta
FTB
t
if
i
3. (SEP5 – Math) Derive an equation for the acceleration of the hanging mass in terms of mh, mt, FTB,
and 𝜇 k. 𝜇 k is the coefficient of kinetic friction between the table and the block on the table.
SYSTEMFnetsys
F E Fgn Masa
FEMME M 9 Ms's
Fa Mama mug Metmala
Fett
Fruth
4
Finna
Mt
B
Solvefor F
F Mna mug
MetMn a FtB Mug
i
Mta Ft Mnatmag Ff
MATE Mna Mag Ff
F Ff
A
3
mÉmn
n
g
meatMna Fa Ming Ming
4. (SEP6 – Explanations) A 294 N object sits on a bathroom scale in an elevator. If the scale reads
400 N, which of the following TWO statements could be true?
400N
a. The elevator is moving up and getting faster.
b. The elevator is moving up and getting slower.
c. The elevator is moving down and getting faster.
d. The elevator is moving down and getting slower.
Explain your answer choice and include a force diagram.
O
TFN
O
IFg 294
Fnetis up b c Fn Fg
IfFayis up so isaccel
If accelis up vetisdown it willdecreasespeed
If accel
also
it willincreasespeed
up tovetis
up an acceleration vs. time
5. On theisgraph
the right, draw
graph for each elevator.
saga
A
a
t
Mr. Fitz challenges his neighbor to a tug-of-war with tractors. A chain
connects the tractors to one another. Mr. Fitz has a tiny Farmall Model A
tractor while his neighbor has a big Case International tractor. The dude just
destroys Mr. Fitz and both tractors accelerate to the left over the line in 3
seconds.
6. (SEP6 – Explanations) How do the magnitudes of the forces that the tractors exert on each other
compare while they were accelerating to the left?
a. Big tractor exerted a larger force on little tractor than the little tractor exerted on the big one.
b. The big tractor exerted a smaller force on the little tractor than the little one exerted on the big one.
c. The big tractor exerted the same force on the little tractor as the little one exerted on the big one.
d. It’s impossible to tell without more information.
o
N 3rd Law
7. (SEP6 – Explanations) Mr. Fitz is driving his tractor at a speed of 5 m/s along a level road when a
deer jumps out in front of him. He slams on his brakes and slides to a stop. The tractor doesn’t stop
instantly. What is causing the tractor to keep moving and to not stop right away?
Thetractor is moving To stop it must accelerate It takes
time to change speed Thetractorhasinertiawhich means it resist
a change in velocity
8. You push your lawnmower (mass = 15 kg) across the lawn
at constant speed. To do so, you must exert 120 N of force
towards the mower at a 35° angle with the horizontal.
a. (SEP4 – Graphs) In the space under the figure, sketch a
force diagram to represent this situation.
b. (SEP5 – Math) Calculate the magnitude of the normal force.
Fnety FN
Jin
Fg Fpy O
Fn Fg Fpy
ftp.fgffit
159.8 1205 35 215.8N
Fn
c. (SEP6 – Explanations) Suppose you pulled instead of pushing, but still at the same angle.
1. What would happen to theco
magnitude of F ?
g
ÉfÉg
Increase
Decrease
Stays the Same
Ft
2. What would happen to the magnitude of F ?
Increase
Decrease
Stays the Same
d. Explain, using words not numbers, how and why the normal force and the force of gravity would
or would not change if you pulled upwards on the lawnmower handle instead of pushing on it.
N
Fg same ble m I g arethe same
ratherthandown Fnety 0 for
Fnhas bereallybigWhenFpyisup
bleFgisthesamebutnowFpyis up
Fn smallerscenarios
down
when
to
both
So
Fndoes nothavetobe
Fpyis
big
9. Chili the cat is playing with a block on a rough inclined plane. He
paws it and it slides up the inclined plane 1 m before stopping.
a. (SEP 4 - Graphs) Draw an FBD for the block as it slides up the
inclined plane (after he hit it, but before it stops).
b. (SEP 5 - Math) Derive an equation for the acceleration of the block
in terms of its mass M, the angle of the inclined plane θ, the
coefficient of friction μ, and other physics constants.
Fnetx Ff atFox Mma
Fray Fn Fgy 0
MEN Fgsince Ma
cost MgsinO Ma
M KMg
MKgcoso gsi.no
Fn Fgy Fgcoso Mgcos0
É
FK ÉÉÉ
10. (SEP4 – Graphs) Two teams conducted an
experiment similar to the one you did (Forces
and Motion Lab) and then graphed their data on
the same graph, as seen at right. The
experiment had them changing the net force
and recording the resulting acceleration. What
did the dashed-line team do differently
compared to the solid-line team? Explain how
you know. You must reference an equation and
connect it to the graph
y
a
mxtb
Fna
Fnet ma
a Emet
att Fat
Slope mats
so
groupuseda
smallermass
11. Betsy Joy the puppy pushes square ice cubes against
each other across the kitchen floor (no friction!). She
applies a constant force F on the first ice cube of mass
3M, which is in contact with the second ice cube of
mass M.
a. (SEP4 - Graphs) Draw FBDS for each block.
Mark any forces and are equivalent.
F
5
41
Fog
b. (SEP5 - Math) Write an equation for the net force on
the small block in terms of the applied force F.
Fast F Msysa 7 F 4 ma
jeg
Em a
c. (SEP5 - Math) Write an equation for the net force on
the large block in terms of the applied force F.
Fnet3m 3Ma
3MEm
3
d. (SEP6 - Explanations) Identify ALL third law pairs in your FBDs.
Only Formby3m andFon3MbyM
a m
Fretonm
FmMt Fnetonm
F
4 Fon
Weight vs. Mass Lab
Goal: To graphically and mathematically model the relationship between the mass of an object and
its weight.
Quantity
MASS
WEIGHT
Vector
or
Variable Unit
Scalar?
Definition
how much stuff is in
an object
scalar
m
force on anobjectdue to the Vector Fg
the
gravitational attraction between Earth
andtheobject
kg
N
Here’s a song that will help you remember the difference between the two: http://goo.gl/b65Y9l
Use the space below to record the data you collect, graphs you make, and conclusions and equations
from the class Board Meeting. Clearly label each part of the space below with what you put in it.
Weight a
y Ax B
Fg
Gravitational
Weight
Mass
Fg
FieldStrength
98 m
Gean
normwalitteras
mass
9.84kg
Fg mg
downwards
WS 1 – Qualitative Force Diagrams
In each of the following situations, represent the object with a dot. Show all the forces acting on the
object, making the length of each vector represent the magnitude of the force. Then label each force using
on-by notation (on what? by what?). Show how the forces balance using congruency marks. Don’t forget
net force equations.
1. The box sits motionless on the floor.
yFngybfloor
jpg
Fy Fn Fg on
3. The crate is suspended from the ceiling by two
ropes.
2. A crate is suspended from the ceiling.
th
system'd
D
Fy
F Fg ON
on box
FgbyEarth
4. Static friction prevents the box from sliding on the
incline.
Fxs Ft Fa ON
i
systemcrate
Fg
ON Spy
Thesetwoare balancingright
Thenwhatis balancingoutgravity
Motivatecomponent
it
system box
say
5. The TV is motionless, hanging by two wires.
6. The brick is motionless. Assume no friction.
I
Lex
Fifty.gg
TV
Is
system
Fx Fix Faxon egg'ath
7. The brick is motionless. Assume no friction.
Howcan
my
systembrick
FghÉI
Yre
EFy Fay g son
Fabytarif
EF
FtFgFONÉg s
Fy tf Fg ON
is
fgggbg.gg
8. The heavy crate is pulled horizontal by a person with
a force that is parallel to the floor. The crate doesn’t
have move.
You
systembrick
iii
with y theforce
iii
systemcrate
Fight
EFF Fx ON
_FpfyÉte
Lex
Fy Fv Fgyon fgggby.gg
Hgogtatin
9. The person gets a horse, and now the horse tries to 10. The box is pushed by a person. It doesn’t move.
pull the crate. It still doesn’t move.
systemcrate
Ex
i
Whereisthis
HowdoesFnand
Frycompare
i
EFystFntFpyFg
Whathappens if theangledecreases
4omiÉÉdfromstatoclass
angle
.
Fn
I
systembox
ONqpgs f
ie
tF ON
Fy Fn FprFg ON
Howdoesthe F comparetothe
one
Assumesamecrate
Fninthelast
WS 2 – Quantitative Force Diagrams
50mintotal toWB
1. The box has a mass of 5 kg and is pushed by a person exerting force of 50 N applied downward at
an angle of 37° above the horizontal. The object doesn’t move. Calculate the value of the following
forces. Need help? Here’s a video: https://goo.gl/JjzrMT
a. Fg
Floor
b. Ff
c. FN
FNEy
Fy Fg ON
EFF
9.8 5kg O
Fv Jonsin370
F 79
y.si
fTFsonbox
byfloor
EF Fp tf O
150Ncos379
4
systembox
FYI
0
pg
Fg
b
49N
FytFgtFn o
2. A person pulls on a 50 kg desk with a 200 N force acting at 30°
angle above the horizontal. The desk doesn’t budge.
Calculate the value of:
i. Fg
It's
ii. FN
iii. Ff
er
Fry Fs
desk Whatwouldbe
system
Fy F Fp2 0
F
different
if it
200170530 0
fi
3. A box of donuts is in peril! You must calculate the correct forces
of tension on the wires to save the donuts! The mass of the box
of donuts is 0.8 kg and θ=55°.
EFyfgtf.jo EFF
x
998 0814
55
sin o
Ey'd
9.5714105
III g
E ftp.gs 9
9
sI 1
θ
onbox
17 0
4
by
ftp.gnyboge z
n
b
Howwouldyourworkchange
ifyoudid É
fgyyyy.gg
Systembox
4. A 100 N box sits motionless on a ramp.
Thisiswhere I taught
a. Draw an FBD.
b. Calculate the value of the normal force.
theangleontheramp
c. Calculate the value of the force of friction.
Fy Fn Fg ON
tf
Eq
20°
Hgs98N
Fxs Fg Fg ON
Ff 100N sin20 ON
100Ncos200 ON
Fn 94N ON
whydowesplitFginto
theramp
BINup
Fw 94Na
components
and
Whynot
5. Determine tension in cable #1
and the tension in cable #2.
#1
I
107.8mn
hd
getting
I
#2
Bothmasses
figgyboothmasses
6. Calculate the magnitude of the
force of tension in each cable.
g
FgÉgÉal
Just4kgmass
107.8Nb
egg
Fi Fiz
5 kg
2
wouldchange
ballwhat
system5kg
if
onsabk
thatpartof
theceiling
yguy
og
w ires
washigherthe
7. A block of bricks is hung over an alley between two buildings. If
the mass of the bricks is 40 kg, how much force are the wires
exerting along the direction of the wires. Both angles marked
are 25°.
171
ftpfi
b
system
Iggy48,8 539.21h
39g y
Iggy 392N
s
Howis it balanced ifthere
are 2 goingup w 464N
andonly one goingdown
w 392N
196N
sin 259
49,61
171 4
6392
8. Correct the student’s statement below.
The weight of the dog is 40 kg, but the gravity of the dog is 400 N
WS 3 – Friction and Angles
1. A 1.5 kg box sits at rest on an incline, despite a force of 5 N pulling it up the
ramp.
a. Draw an FBD.
b. Calculate the magnitude of the friction force.
HowwouldtheFBDchange
c. Determine the direction of the friction force.
if we stoppedpullingbut
thebox stilldidn'tmove
EEs Fytf t Fg ON
system box
5N 1 5 15kg 9.8 Sin Is ON
g
Fg 1.2N
F 1.2N
v:
FN
Y
a:
Fnet:
i
F is downtheramp
since wealreadyputin
a signforfrictionthe
negtellsus our guesswas
Draw arrows to
represent direction
of:
So f mustbe
wry Fox 3.8N inthesamedirection
as
g
Youcan't calculateMs bc youdon'tknow if thisis Fsm ornot
2. Some masses are added to the box in the previous problem, so now the mass is 3 kg. The same
force is still applied as before up the ramp. The box remains at rest.
a. Draw an FBD.
b. Calculate the magnitude of the friction force.
c. Determine the direction of the friction force.
system box
Draw arrows to
represent direction
of:
FN
Y
EE FgFg Fg ON
5N 1 9 3kg9.8 Sin Is ON
g
F 2.6N
F s 2.6N Fs is up therap
since wealreadyputin
a signforfrictionthe
pjstellsus our guesswasright
v:
a:
Fnet:
FP 5N
pg s y 6
SoFmustbe
direction
inasthegopp
Is
frictionalwaysopposite a pull
force Howdowe figureit out
3. You are pushing a box of rabid dogs down the hallway in order to get them contained in a
classroom. You exert a horizontal force of 150 N on the box and the box moves at a constant
velocity down the hallway. The dogs and box have a combined mass of 64 kg.
a. What is the coefficient of kinetic friction between the box and the floor?
b. If we kept the mass of the box the same but made it have a surface area that is 1.5 times
greater, by what factor would this change the coefficient of friction?
Draw arrows to
represent
direction of:
My 12kg5 60kg
Yw
Mb 4kg
v:
M 64kg
Zero
Fnet:
zero
AreFNandFgalwaysequal
whatistheonlythingthat
a:
6272
EFTson
canchangethe M
Fg
627.2N
FK Ma Fn
150N Ma627.2N
MK O.LY
4. A boy is pushing his younger brother, who has a weight of 250 N,
in a box across the yard with a force of 300 N. The coefficient of
kinetic friction between the cardboard box and the grass is 0.35.
The angle his arm makes with the horizontal (as marked in the
diagram) is 50°.
a. Draw and label a force diagram for the box.
b. Calculate the net force on the box.
Draw arrows to
represent direction
of:
v:
a:
Fnet:
s
qq.tn rjitg o
F 300Nsinso 250NON
F48O
whyisn't
t
puffy
3005150 Ma F
I
192.8N 0.351480N
EFy4
c. How would the magnitude of the net force change in the previous problem if the boy
PULLED with the exact same force and angle?
____Smaller net force ____Same size net force
____Larger net force
Clearly justify your answer choice with words.
_____________________________________________________________________________________________________________
in Fn
_____________________________________________________________________________________________________________
would decrease theFw A
Pulling
up
_____________________________________________________________________________________________________________
because
Fic Matt Less Fk means a smallerface opposingFpx so
in Fk
direction
decrease
net
means a decrease
the
forcein
x
_____________________________________________________________________________________________________________
a larger
_____________________________________________________________________________________________________________
5. Working with a bear conservation group, you tagged a tranquilized 364 kg
grizzly bear. Now you need to get the bear out of the facility before he
wakes up. µs between the bear and the ground is 0.38 and µk is 0.27. You
can pull horizontally with 534 N. Can you move the bear on your own? If
not, how many people total (including you) with your strength will you need to get the bear
moving?
withoneperson
Draw arrows to
represent direction
of:
zero
a:
zero
zero
with3people
I
FindFs
max
F
v:
Fnet:
M s 0.38
3,567.2N
bearmoving
Mw
toget
13.56720
I
canexert
2.5peopletotal
IIIIIwill
tabeatL
thanthis
ftp.T
the
gum
f
3peop6t
Fg 3,567.2N
What isthedifference between
Fand u
Whatwillhappenwhen 3peoplestart pulling
6. Box Pulled on a Rough Horizontal Surface – Frictional Force on Box (nT5H-CCT94)
A 100 N box is initially at rest on a rough horizontal surface. The coefficient of static friction is 0.6 and
the coefficient of kinetic friction is 0.4. A constant 35 N horizontal force to the right is applied to the box
as shown. Several students are discussing the frictional force exerted on the box
by the rough surface 1 second after the force is first applied:
Al:
“The frictional force is 60 N since the box will not be moving and the
coefficient of static friction is 0.6 with a normal force of 100 N.”
Brianna: “The frictional force is 40 N since the coefficient of kinetic friction is 0.4 and there is a normal
force of 100 N.”
Carlos:
“The frictional force is 35 N since the box will not be moving and the frictional force will
cancel out the applied force of 35 N.”
548
David: “It is 40 N for the kinetic frictional force and 60 N for the static frictional force. The normal force
is 100 N and the coefficient of kinetic friction is 0.4 giving 40 N for the kinetic friction. Similarly,
for the static frictional force it is 60 N since it has a coefficient of static friction of 0.6.”
Which, if any, of these students do you think is right?
Al
Brianna
Explain your reasoning.
Carlos
0
David
None of them
Staticfrictionwilladjusttoperfectlybalanceout
opposingforces
up to its max
any
value whichin thiscase is up to 60Nwhichislessthantheapplied35N
WS 4 – Newton’s 2nd Law
For each of these problems, you must begin your solution with a force diagram, direction arrows,
and/or a picture.
1. The maximum force that a grocery 2.
bag can withstand without ripping
is 250 N. Suppose that the bag is
filled with 20 kg of groceries and
lifted with an acceleration of 5 m/s2.
Do the groceries stay in the bag?
Start with and FBD.
system bag groceries
IF
big196N
Fp 296N
IEEE
Doesitbreak Fp250NbutFive 250N
Whatcausessomethingtobreak
a netforce or a force
Whatforcemakesthecar
goforward
M 1.000kg what wedidn't know
if
Vo255
themass thecar
Fff
g Eggen
196N
system car
p
stg
a
Suppose you are driving a 1,000 kg car is traveling at 25
m/s (≈55 mph). When you slam on the brakes, the
maximum frictional force between the tires and the road is
5,000 N. What is the minimum distance required for the car
to skid stop? Start with an FBD. Hint: Find acceleration,
m
then find ∆x.
v o
Fa 5,000N
ÉÉ
V2 Lasxtv
105 2154 agita
03 265511 1253
giggly
q
of
1
62.542
53
3. During a head-on collision, a car accelerates from 13.3 m/s (≈ 30 miles/hour) to rest in 0.10 s.
a. Assuming the passenger is wearing a seatbelt, and thus stays with the car, what is the
acceleration of the passenger?
b. If the passenger isn’t wearing a seatbelt, what will happen and why?
c. The driver of the car holds out her arm to keep her 25 kg (55 lb) 5 year old child from
smashing into the dashboard because the kid isn’t wearing a seatbelt. What force must she
exert on the child to prevent a catastrophe?
d. Convert these forces from newtons to pounds. (1 lb = 4.4 N). What are the chances the
driver will be able to stop the child?
IFN
systempassenger
Fseatbelt
8
a v attro
03 a 0105 13.35
9 1337
c
a EyedEmt
1337
kg
d 3,32514444
756lbs
Whatforcemakesthepassenger
goforward
3,325N whatforceisthis
b
The passenger would continue moving forward aka "fly forward" into
the dash. This is because of Newton's 1st law: the passenger is moving
with a constant velocity and since he is not wearing a seatbelt, no net
force is acting on him, so he will continue to move at a constant
velocity. The car stops due a force from the other car it crashed into.
4. The system at right is at rest. If the ball weighs 8 N,
a. Start with and FBD for the junction of strings.
b. What are the tensions in the ropes?
c. What is the magnitude of the friction force between the block
and the table? Draw an FBD for the block.
d. If the block has a mass of 5 kg, what is the coefficient of
friction between the block and the table assume that the block
experiences the maximum force of static friction?
Fy Fgtfasin1307 ON b
a
8N Fasir3070N
E
EE Fi tf cos30 ON
Fi 16Ncos30 ON
FF
13
Howdodealw
out
f
o
13.9N
F 13.9
misfit.ie98
Ms 0
IIfIEEIjj
Is thistheusthough Doweknowfrictiongb
this isthemaxstatic
5. An 840 N student is being pulled up by a rope with a force of 1050 N. The force drops to 840 N
and then the force drops to 588 N, all for equal amounts of time.
a. Draw an FBD for each scenario and describe what is happening to the speed.
1050 N Force &
moving up:
What is happening
to the speed?
I 1,050N
Ffg 840N
588 N Force &
moving up:
What is happening
to the speed?
Fg 840N
b. Calculate the acceleration at the beginning and end of the trip.
A Fff
4,4
840N M 9.87g
1.058
548 2.4545 A
588N
M
Fff 58845544 2.945
c. Complete the graphs below for the student during the entire time they are pulled as
described above.
ay
y
y
vy
t
pagan
t
t
Fnet
FT
FETTLE
t
Mr
t
WS 5 - Intro to Two Body Systems: Atwood’s Machine
1. Go on the class website and watch the video tutorial.
MF
fye
t
system bothblocks
System maonly
Eidmagadaation
Findtension
light EF
Fg
Fga Fg
m tma a
Mag mig
s m tma a
79min11s a
EFa maa
Fga Ft
maa
Mag Ft Maa
s
Ft Mag Maa
Big idea questions:
1. How does the acceleration of m1 compare to 6. To find acceleration is it best to focus on one
individual object? Or the system as a whole?
the acceleration of m2?
Explain.
same
2. How does the tension that m1 feels compare to
the tension m2 feels?
game
Fi Fg
thetensionforcesbetween
eliminates
themasses
oppositedirections
3. How does FT on m1 compare to the Fg of m1?
Systemas awholeDoingthis
7. To find FTension is it best to focus on one
individual object? Or the system as a
whole? Explain.
4. How does FT on m2 compare to the Fg of m2?
Fists
5. Why do we bend the coordinate system?
Otherwiseour systemwould
bemovingin two directions at once
Individualobjectbecausethere
aren'ttensions on thesystem
as a whole
2. Assume there is no friction on m1.
a) Bend your coordinate system and choose a positive direction.
b) Draw the FBDs for this system. Label forces with +/-.
c) Solve for acceleration symbolically.
d) Solve for acceleration assuming m1 = 3 kg and m2 = 2 kg
e) Find FTension acting on m1
f) Find FTension acting on m2
Sys
s
m tmz
II Iim MIT
3.92
d as ygK9j
I
0
6 Asf
f
3.927 38
Éittgg
98
7
3431
3.9252
Filling
big
WS 6 – Elevators
1. A physics teacher is an elevator that is
moving downwards and is slowing down.
Enter ↑,↓, or zero for the velocity,
acceleration, and net force on the physics
teacher. State the relation between the
normal force from the floor of the elevator on
teacher and the force due to gravity acting on
the teacher. (i.e., FN = Fg, FN > Fg, or FN <
Fg). Draw the FBD.
v
2. A physics teacher is in an elevator that is
moving upwards at a constant speed.
Enter ↑,↓, or zero for the velocity,
acceleration, and net force on the physics
teacher. State the relation between the
normal force from the floor of the
elevator on teacher and the force due to
gravity acting on the teacher (i.e., FN = Fg,
FN > Fg, or FN < Fg). Draw the FBD.
v
v
T
a a
a zero
Fnet
Fnet zero
3. Match the following:
FN > Fg
Fnet = 0
FN < Fg
Fnet ↑
FN = Fg
Fnet ↓
4. A 100 kg guy is having a phun Halloween
riding elevators. Rank the scale readings
from largest to smallest.
I
g
IIe ÉE
Far Fg
intaggonatfersofecyweighess
5. A woman stands on a scale in a moving elevator. Her mass is 50.0 kg, and the combined mass
of the elevator and scale is an additional 700 kg. Starting from rest, the elevator accelerates
upward. During the acceleration, there is a tension of 9000 N in the hoisting cable.
(a)
What is the acceleration of the elevator (and the woman and scale inside of it)?
(b)
What is the reading on the scale during the acceleration?
up
A f
Met I
woman
AFN
t
woman
b
Fnet
Ft Fy Ima
elevator
andwoman
50119
OFgygon Fg
Fg
750 98
pg1350N
FBDsnotdrawnstale
9000 7350
750a
Fnet ma
Fn Fg ma
2
Fn 490 50712
Fn 600N
with at
FBI
af
FBP with
Top
6. A woman is in an elevator watching the
scale reading of a hanging fish. Physics
is phun. The elevator is accelerating veg
upward in the diagram on the left, and
it is accelerating downward in the
diagram on the right.
asks
Foto m
If the fish weighs 40 N, and if the magnitude
of the acceleration is +/- 2 m/s/s,
determine the scale reading while the
elevator is
(a) accelerating upward and
(b) downward.
b
a Fnet Ma
Ft Fg ma
Fnet
ma
Ft Fg ma
4.0871 2
40
Ft
F 40 408 2
F 48.2N
Ft 31.8N
7. The graph below was collected throughout an entire elevator ride of your phriend. The data
collected was the upwards force on the phriend. During each segment…
this data
so
actually
a. Label relationship between FN and Fg (ie. FN > Fg)
elevator
normal
initiallyb. Describe the motion (both direction AND what is happening to speed)
of
acting
is
force
arrest
p
gjII
T
speeding
moving
up
downwards
Ecg
É
ingg
qq.gg
p
I Fg
1kg
a
on
JEFF
r
wtf p hi ÉYga
FI
a8
ye
8. A physics student riding in an elevator is slowing down while arriving to the third floor from
the first floor. Do they feel lighter, heavier, or normal?
Picture & FBD
Equations and/or physics concepts:
v f
Enett
Fn
gvFg
Select Evidence (Given This…)
Giventhattheelevator is slowing
down
Fnet ma
Fn Fg ml a
see v a
Fnetbox a
is directed
downward
Reasoning (Means that…)
acceleration vector
thatthe
p This meansopposite
thevelocity vector
is in the
vector representing
Given that Fnet ma
Claim:
This means thatthedirected downwards
the netforce is also
andFn Fg
Since the floor exerts a smaller force on the student upwards
then Fg on thestudent downwards the studentfeels lighter
WS 7 – Two Body Problems: Atwood’s and Flatwoods
9. At right is a picture of an Atwood's Machine: two masses attached to a pulley.
The mass of block A is 5 kg, and the mass of B is 2 kg.
a. Draw the necessary force diagrams.
b. Calculate the acceleration of the system when the blocks are released.
c. Calculate the tension in the string after they are released.
d. How long will it take for block A to fall 2 m?
b Asys
Fsg ftp mg C8EfgKf9521d 4.22
BlockA a
BlockB
Eg ftp
a
It
It
d by fatVogt
I i
4
9
1171
818141 4.23 28102
iffy
2m214.25t
4.23
1
0.980
É
F 28N
propatiing
b asys 5515311 3mg 3g asys
10 Block A requires
whyaddthemasses
couldthisever bein freefall
ofthenetforce
onthesystemtogetthesameaccel
ofthesysble a Eventand ithas5of
themass 3mg
ingg 21Nfive
Even Fg F 21N5mg
F
I'd
10. A block is being pulled up a ramp as shown in the diagram below. The coefficient of friction
between the ramp and the block is 0.5.
a. Draw the force diagram for the block on the ramp.
b. What is the normal force acting on the block?
c. What is the acceleration of the block?
d. What would the acceleration of the block be if it was
pulled down the ramp?
b EFy Fn Igy ON
forces helpinghurting
What
are
not
and directlyaffectingaccel
FN Mgcos o ON
Fr 15kg9.8kgcosts on
15
120.4120
Fg F 150N ma
120.4N 150N
sky a
15kg 9.8kg sin350 o5
C EE
84.3N 60.2N 150N 15kg a
5.5N sky a
EE
9
0.3732
iffy
d FrictionandFpswitchsigns
84.3N160.2N150N 15kg a
EF 174 N Csky a
ll.tn
as
11. Suppose a 10 N hanging mass (mass B) is attached to a 40 N
A
block on the table. The coefficient of kinetic friction
between the block and the table is 0.20.
a. Draw your force diagrams.
B
ga
b. Will the force of tension in the cord be the same
magnitude as the force of gravity on block B before the
system is released from rest? Why or why not?
c. Will the force of tension in the cord be the same
magnitude as the force of gravity on block B after the
system is released from rest? Why or why not?
d. If the coefficient of kinetic friction, µk, is 0.20, what is the acceleration of the block?
e. What would be the minimum value of the coefficient of static friction, µs, in order for the block
to remain motionless?
f. Calculate the tension in the cord while the system accelerates.
É
b Yesbecausebefore it isreleased it isnotacceleratingThis
meansthenetforcemustbeON Sincethereare onlytwo
forcesactingonthemass
e
Fg Ft
EntireSystem
EF
c Nosince it isacceleratingthenetforcemustbenonzero
sincethereare onlytwoforcesactingonthemass Fg
d ÉÉ
Fj Fgg
um g tmag m t ma a
0.207140N t ION
5.1kg a
as 397,1mA
asys
8 N t 10N 5.1kg a
9
0.39272
Fy t Fgg ON
Mm g tMag ON
M 40N ION ON
Ms 0
Ft
d ProportionalReasoning
mamaa
fit
f asFYI FIFI
0.3925
mgjo8mg
IFt9
879 o.oygso.gg
12. The disembodied hand exerts a 5 N force on the 100 g mass. Friction between the boxes and the
floor is negligible.
a. Draw an FBD for the system, for A, and for B.
A
B
b. What is the acceleration of each box? Draw
100
g
Acceleration isthesameforboth
Attest
8 kg 3.03
Both
boxes
i
É
50
g
WS 8 – Two Body Problems: Bricks and Blocks
1. Suppose m2 is four times that of m1. If the friction between the
cart and the table is negligible, what is the acceleration of the
system, in terms of g? Draw an FBD for each block.
anting.immim
iA sit
neededtoaccel
5theFnet
8
Any
Whatif therewasfriction
As 4mgImI
dd
t
whatwouldchange if thetablewas aramp
KT
2. Now supposed m1 and m2 are switched, so m1 is the hanging mass and m1 is the mass on the
table. What is the acceleration of the system, in terms of g? Explain any differences in the
acceleration of the system, even though the total mass of the system remained constant. Draw
an FBD for each block.
45theFnetneededtoaccel
a tag.mnmimtT
How dothe accelerationscompare
4mm Ysthe
Fnetneededto
accel
Whyisn'ttheaccel9.87iftherearenoresistiveforces
3. The person pushes box A with a force of 16 N. Friction is
negligible.
a. Draw FBDs for the system, A, and B.
b. What is the net force on the system?
c. What is the acceleration of the system? Box A? Box B?
d. What is the net force on Box A?
e. What is the net force on Box B?
f. How much force does Box B exert on Box A?
g. How much force does Box A exert on Box B?
b 16N
Lookatwholesystem
c
a Emet
d Feta Ma a 3kg4m15 12N
EÉfTun Biffegg
Lawpains
Whichforcesare3rd
ABsyscope
theaccelof
Howdo
Fret
d Prequiestofthetirefthesystem e Brequires theFiveofthesystem
43.16N 12N
ftp
g 4N 3rdLaw
1kg4ms 4N
doesinordertoaccelerate
B
1
kg
f Samea Fneta 4N
É g43 forall
e Five
A
3 kg
inordertoacceleratewith it
SoFm 16N 4N
4. The person pulls the string attached to Box B with a
force of F. Box A and B are connected by hooks.
A
Friction is negligible. Give your answers in terms of
100 g
multiples/fractions of F.
a. Draw FBDs for the system, A, and B.
b. What is the net force on the system?
2 3 thesystems
c. What is the net force on Box A?
d. What is the net force on Box B?
mass is here
e. How much force does Box A exert on Box B?
f. How much force does Box B exert on Box A?
B
50 g
of
Bothhavethesame acceleration b c they are
connected I dividually
theyhavedifferent Fnet's
b Fbythehandsince
becausetheyhave different mass
thereisnofriction
3F
c
Yoofthesys mas
A
d IF
3F
s netforce is 2
because it has 2 3
mass
ofhastheto
mustbeYs It
F the
Fuel Bso
the
be
F
must
to
to
left
43
right
2
F BA give13 a FnetofYzfThe 37 tothe
is a3rdLawpain
A
IF
f 3F
e
left
FIEF by a
on
Hand
5. Boxes are pulled by ropes
along frictionless surfaces,
accelerating toward the left.
All of the boxes are identical,
and the accelerations of all
three systems are the same.
D
B
E
A
C
F
6. Rocks that are thrown up into the air all have the same shape, but they have different masses.
The masses of the rocks and their speeds when they were thrown are given.
C
B
A
D
WS 9 – Two Body Derivations
1. At right is a picture of an Atwood's Machine: two masses attached to a pulley. The mass
of block A is 5M, and the mass of B is 2M.
a. Draw the necessary force diagrams.
b. Derive an equation for the acceleration of the system when the blocks are released
in terms of M and g.
c. Derive an equation for the tension in the string in terms of g and M.
b Fret
A
Iff
ff
Fga
Fnet ma
FB Fg Mpa
ft 2mg 2M3g
ftp 2mg Mg
c
ma
Fga Fgb Mathis
Mag MBG Mathis
5M 2M a
5mg 2mg
ypg y
3mg
Ma
Ftr 2mg
a
3,9
y
FnetA Maa
Mg
Mg
Fga Ft Mad
Ft 5m13
5mg
5mg Mg Ft
My
F
yay
2. A block of mass M is being pulled up a ramp with a force FT as
shown in the diagram below. The coefficient of friction between
the ramp and the block is μ.
a. Derive an equation for the acceleration of the block on the
θ
ramp in terms of Ft, M, θ, g, and μ.
b. Using your derived equation, write an
equationincline
for the acceleration
if there was
as
Used
no pull force (still moving up the ramp).
c. Using your most recent equation, write an equation for the acceleration if there
was no friction (still moving up the ramp).
d. How would the answers to (b) and (c) change if the block was sliding down the
ramp?
up
ft
TY
a Fret ma
Fnet ma
FInfo Fgsint ma
maf.ie
Ft MFycost Fysint ma
E mmgcoso mgsino
Ft mmgcost
d if slidingdown
imagine thisISTE
É
tm
Mgcost gsinga stiffed
stgpna
ma
mgsino.mg's nÉ
Partb frictionwould directed up the incline
Set Ft O and makeFr
c
ohÉÉtgEe
nochange
A
t
same direction as
so Mgcost gsinosa
E
g
g
3. Suppose B has a mass of 3M and A has a mass of 4M. The
coefficient of kinetic friction between the block and the table
is μ.
a. Draw your force diagrams.
b. Derive an equation for the acceleration of block A in terms
of M, μ, and g.
ÉFw
Fft
ÉÉIÉ
If
IFYA
STsA
III
syp
Fret ma
Fret
system
ma
FB Ff Matm a
Fg ft
So Ff umag M 4M g
Ft yFn Maa Mig
Ft MMAg Maa 3mg 4Mat4yMg 3Ma
Ft414Mg 4Ma
B
and for
ma
Ft MBA
Mmda
To
AisalsoaceforB
b accelerationforwhole
Fg
545
A
3 94919 3Mattila
3gjul4
a
3g
4M5_aaFt
4. The person pulls the string attached to Box B with a force
of F. Box A and B are connected by hooks. Box A has a
mass of 2m while Box B has a mass of m. The coefficient
of friction between both blocks and the floor is µ.
4Mat4MMg.eftp
A
B
Give your answers in terms of F, m, and physics constants.
a. Draw FBDs for the system of both blocks, just block A, and just block B. Bona
b. What is the net force on the system?
c. What is the net force on Box A?
d. What is the net force on Box B?
e. How much force does Box A exert on Box B?
f. How much force does Box B exert on Box A?
fEii
FFnetp
Fnet13
system
pye
Theffetemphepin's bitty
and
Note
Jj
Frseytstem
F
Yystem
0.10 0.05 a
Fionn
0.10781s
Fiona
FnetB
F
F
FBon a 3
e
they
IF
Faon B
f Fpon A
Mk
F
IF
F Faon
F FAonB
3