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Chapter 4
NEWTON’S LAWS, FORCES, &
FREE-BODY DIAGRAMMING
HONORS PHYSICS, 2014-2015
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In-Class Problem Solving
EXAMPLE 1 (4.6): A freight train has a mass of 1.5 x 107 kg. If the
locomotive can exert a constant pull of 7.5 x 105 N, how long does it
take to increase the speed of the train from rest to 80 km/h?
EXAMPLE 2 (4.14): The force exerted by the wind on the sails of a
sailboat is 390 N north. The water exerts a force of 180 N east. If
the boat (including the crew) has a mass of 270 kg, what are the
magnitude and direction of the acceleration?
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Free-Body Diagramming:
Consider all of the forces acting on a crate being pushed across a
horizontal floor.
How can we calculate the acceleration of the box?
Free-Body Diagramming: 1-D Elevator Example
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Friction FBD Example:
Suppose you shove a box across a surface. What is the acceleration of the box as
it slows to a stop?
CASE A: FLAT SURFACE
CASE B: SLOPED SURFACE
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BOX SLIDING DOWN A RAMP EXAMPLE
What is the acceleration of a box sliding down an inclined ramp a) when neglecting
frictional forces, and b) when considering frictional forces?
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Name: _____________________________
Honors Physics FBD Activity
End-of-the-semester fun with inclined surfaces!!! 
Purpose:
This seemingly simple lab activity will shed some light on the frictional force.
Supplies Needed:
Spring scale, quarter, text book, weight, protractor, and boards (for inclination)
Directions for Part ONE:
(read thoroughly before beginning)
Before beginning, wipe down your book cover with a dampened paper towel to remove
any surface oils, dirt, or adhesives. Dry the cover thoroughly.
Placing a coin on your textbook cover, begin inclining the cover upright slowly.
Have someone measure the angle (try to be as accurate as possible!) as you
continue inclining the cover. I highly recommend you slowly raise the cover
vertically, and then pause for a moment every degree, being as the coin might
finally reach the verge of slipping.
Once the coin slowly starts moving down the slope at what appears to be a constant
velocity, record this angle. Do this multiple times, being as our measurement
tools aren’t tremendously precise. I recommend putting the coin back at the same
initial book cover location. Once your entire group has come to a conclusion
regarding the proper angle (you can use an average, or take the most consistent
result), record this value in the space below.
Angle Measurements (use 2 sig figs): ___________________________
Average angle (2 sig figs): ___________
Equation Derivation for μk:
Derive an expression for the coefficient of kinetic friction by drawing a FBD of
the coin moving at constant speed down the slope. First, draw a diagram of the
situation depicting the textbook, the coin, and the following physical necessities
for Newtonian Mechanics: FW, FN, FF, and θ. You should NOT use any numbers in your
equation derivation, and please keep your angle as theta (don’t plug it in yet!
This expression works for ANY angle, not just this situation!!). Translate your
picture into a rotated coordinate system, and remember that the coin moves at
constant speed down the incline as you derive your expression for the coefficient
of friction between the coin and your book. As a hint, your mathematical
expression for μk should be a trigonometric function!!!
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What seems particularly interesting for the equation you derived from a physical
standpoint?
Compare your coefficient for kinetic friction with that of i) steel on ice and
ii) rubber on concrete(see your book or your equation sheet). At what angle would
expect a steel object to begin sliding down an icy incline?
Compare your angle value from the previous page (the weighted value given to two
significant figures) to some of your classmates recorded values. Explain why
there might be some differences (hopefully minor).
Directions for part TWO:
Once you have μk, now place the weight on your slope at some arbitrary angle. Use
the provided board(s) to incline your book cover. Be sure to measure the angle
made with the horizontal as accurately as possible! Attach your spring scale to
the weight, and slowly continue applying force until the weight begins to slide at
a constant velocity in the uphill direction. Have someone carefully assess the
spring scale force value (in Newtons), and write this value down. Do multiple
trials to check for precision and accuracy. Finally, draw a FBD for this motion,
and solve for the Normal force provided by your book surface and the Frictional
force opposing the motion of your weight. Of course, you will need to use the
friction coefficient from the first part of this lab to do your calculations!!
Show FBD and All Calculations HERE:
Frictional Force: ____________________
Normal Force: ________________________
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Name: _______________________________
Honors Physics In-Class FBD Worksheet
Conceptual Practice
1) Your physics book sits motionless on a perfectly flat tabletop.
Draw a free body diagram of this situation and label all forces
acting on the book.
ΣFx: ________________
ΣFy: ________________
2) Mr. Cunnings is suspended motionless from our ceiling by two
vertical ropes. Draw a free body diagram of this situation and
label all forces acting on poor Mr. Cunnings.
ΣFx: ________________
ΣFy: ________________
3) An egg is freely falling from a nest in a tree. Neglecting air
resistance, draw a free body diagram of all of the forces
acting on the egg as it falls.
ΣFx: ________________
ΣFy: ________________
4) A flying squirrel glides from a tree toward the ground at
constant velocity (ignore flapping of wings). Take air
resistance into account, and label all forces acting on the
squirrel in a free body diagram.
ΣFx: ________________
ΣFy: ________________
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5) Your physics book sits motionless on a perfectly flat tabletop.
You apply a horizontal force that slides the text book with a
rightward acceleration. Draw a free body diagram of this
situation and label all forces acting on the book.
ΣFx: ________________
ΣFy: ________________
6) Your physics book sits motionless on a perfectly flat tabletop.
You apply a horizontal force that slides the text book with a
rightward constant velocity. Draw a free body diagram of this
situation and label all forces acting on the book.
ΣFx: ________________
ΣFy: ________________
7) A college student rests a backpack on her shoulder. The
backpack is suspended motionless by one single strap from one
shoulder. Draw a free body diagram of this situation and label
all forces acting on the backpack.
ΣFx: ________________
ΣFy: ________________
8) Mr. Cunnings goes skydiving (again). He jumps out of the plane
and is accelerating toward Earth’s surface. Draw a free body
diagram of the vertical forces acting on him.
ΣFx: ________________
ΣFy: ________________
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9) During Mr. Cunnings’ skydive, he reaches his “terminal
velocity” and is now falling with constant vertical velocity.
Draw a free body diagram of the vertical forces acting on him.
ΣFx: ________________
ΣFy: ________________
10)
A force is applied to the right to drag a sled across a
flat surface of loosely packed snow with rightward
acceleration. Draw a free body diagram of all forces acting on
the sled.
ΣFx: ________________
ΣFy: ________________
11)
A force is applied to drag a sled up a hill inclined 23
degrees with rightward acceleration parallel to the hill slope.
Draw a free body diagram (original AND translated) of all
forces acting on the sled.
ΣFx: ________________
ΣFy: ________________
12)
A force is applied to drag a sled up a hill inclined 15
degrees with a rightward acceleration. The applied force is
administered at an angle that’s 35 degrees northerly with
respect to the hill slope. Draw a free body diagram (original
AND translated) of all forces acting on the sled.
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ΣFx: ________________
ΣFy: _______________
13)
A block of mass m on an inclined plane is joined to a mass
n by a cord over a pulley, as shown below. Draw translated
free body diagrams for both blocks.
ΣFx: ________________
ΣFy: ________________
14)
Two blocks of mass m1 and m2 are connected by a light
string passing over a pulley. The blocks are at rest on
inclined frictionless surfaces, and the effects of the pulley
are negligible. Given the picture below, draw translated free
body diagrams for both blocks! For this problem, you will need
to use subscripts for all forces acting on each of the two
blocks (e.g. FN1).
ΣFx: ________________
ΣFy: ________________
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15)
A lantern of mass m is suspended by a string that is
joined to two other strings, as shown in the diagram below.
a. Draw a free body diagram for the forces acting on the
lantern.
ΣFx: ________________
ΣFy: ________________
b. Draw a free body diagram for the forces acting on the
knot.
ΣFx: ________________
ΣFy: ________________
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Name: ___________________________ Date: ________ Score: ___________
Honors Physics
Newton’s Laws: Free Body Diagramming
Equilibrium Under the Action of Concurrent Forces
Definitions:
Concurrent Forces: Forces whose lines of action all pass through a
common point. For concurrent forces, we will treat all objects
(e.g. boxes, humans, footballs) as point objects, which greatly
simplifies the problems.
Equilibrium: An object is said to be in equilibrium under the action
of concurrent forces provided it is not accelerating. For all of
the problems on this worksheet, the objects are in equilibrium,
meaning that they are either not moving OR that they are moving with
a constant nonzero velocity.
Weight (Fw):
object.
The force with which gravity pulls downward on an
The Tensile Force (FT): The force acting on a string, cable or chain
(or any structural member, for that matter). The tensile force will
always tend to stretch these objects. The scalar magnitude of the
tensile force (e.g. 22.4 N) is called the tension.
The Friction Force (Ff): The force that opposes the sliding of an
object across an adjacent surface that it is in contact with. The
friction force will be parallel to the surface and apposite to the
direction of the motion (or impending motion) of an object.
The Normal Force (FN): The force on an object that is being supported
by a surface. The normal force will always be perpendicular to the
surface supporting the object.
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Directions:
Complete the following problems. Give your final answers with the
indicated number of significant digits.
1)
In the figure shown, the tension in the horizontal cord is
30.0 N. Find the weight of the hanging object. Use three
significant figures in your final answer.
2)
A rope extends between two poles. A 90.0 N boy hangs from the
rope, as shown in the diagram. Find the tension in the two
parts of the rope. Use two significant figures in your final
answers.
FT1: _______________
FT2: _____________
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3)
A 50.0 N box is slid straight across the floor at a constant
speed by a force of 25 N, as shown in the figure. Use two
significant figures for all answers.
a. How large a friction force impedes the motion of the box?
b. How large is the normal force?
c. Find the coefficient of kinetic friction between the box
and the floor
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4)
Find the tensions in all of the ropes shown in the figure,
assuming that the supported object weighs 600 N (use two
significant figures for all answers in this problem).
Answers:
5)
a)
b)
c)
FT1: ______
FT2: ____
FT3: ____
FT4: ____
FT5: _____
Each of the objects shown in the figure is in equilibrium.
Find the normal force, FN, in each of the three cases. Use 1
significant figure for a and c, and 2 significant figures for
b.
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6)
Considering the same three cases in number 5, find the
coefficient of kinetic friction if the object is moving with
constant speed. Round all of your answers to two significant
figures. Use two significant figures for all answers.
7)
Pulled by the 8.0 N block
block slides to the right
between the block and the
frictionless, and use two
a)
b)
c)
shown in the diagram, the 20 N
at a constant velocity. Find μk
table. Assume the pulley to be
significant figures in your answer.
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Name: ____________________________ Date: ________ Score: __________
FBD Applied Problems (no diagram provided)
1)
Rudolph, the red-nosed reindeer, pulls on a rope attached to
a 20.0kg sleigh (Let’s assume Santa isn’t in the sleigh yet,
nor are the presents for all of the children of the world).
Rudolph pulls with a force of 90.0N at an angle of 30.0
degrees to the horizontal. The coefficient of kinetic
friction between the sleigh and the snowy surface is 0.200.
a. Draw a free-body diagram of this situation
b. Draw a vector diagram of this situation
c. Find the acceleration of the empty sleigh.
\
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1) Suppose that a box of mass m hangs by a string in an elevator that’s accelerating upward, as shown in
the image above. Using a free body diagram (drawn to the side of the image), derive an expression for
FT for this situation.
a. How would your expression change if the elevator WAS NOT ACCELERATING?
b. How would your expression change if the elevator were accelerating DOWNWARD?
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2) Now let’s use some numbers. If the box has a mass m = 5 kg and g = 9.8 m/s2,
a. What is the tension in the rope if the elevator is MOTIONLESS?
b. What is the tension in the rope if the elevator is moving at constant velocity?
c. What is the tension in the rope if the elevator is accelerating upwards at 2.0 m/s2?
d. What is the tension in the rope if the elevator is accelerating downwards at -1.5 m/s2?
e. What if the elevator cable snapped and the elevator was freely falling down the elevator
shaft…what would the tension in the cable be?
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3) Reconsider the elevator situation described in problem 2. In other words, the mass of the box is still
5kg. Suppose that the rope connected to the hanging mass can only withstand tensions less than 75
Newtons. Tensions greater than 75 Newtons would cause the rope to snap. What maximal upward
acceleration could the rope withstand before it snaps?
4) Another important equation we’ll encounter that involves springs is F = kx, where F = force, x = the
spring stretch/compression distance, and k is the “spring constant.” Note: This equation –Hooke’s Law
– is sometimes written as F = -kx, being as springs provide a RESTORING FORCE…for now, however,
let’s keep things simple. A box of mass m is hung by a spring from the ceiling of an elevator. When the
elevator is at rest, the length of the spring is L = 1m.
a. As the elevator accelerates upward, the length of the spring will be _______ 1 meter
i. Greater than
ii. Equal to
iii. Less than
iv. It’s impossible to know based on the information given
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b. Now let’s use some numbers in this situation. A single mass m1 = 5 kg hangs from a spring in a
motionless elevator. The spring is extended x = 12 cm from its unstretched length. What is
the spring constant, in N/m?
c. If the elevator accelerates upward at 2.5 m/s2 what will the length of the spring be when
compared to its unstretched length? Give your answer in CENTIMETERS.
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Name: __________________________________
Honors Physics
FBD Retired Quiz 1
1)
Homer Simpson falls off a power plant tower, and, while freely
falling to Earth, two of his hairs get snagged around a utility
wire. One of the snagged hairs (label this hair 1) is oriented 30
degrees to the left of the vertical, and the other hair (label
this hair 2) is oriented 15 degrees above the right horizontal.
Homer’s weight is given as 1000N. Assume that both hairs are
weightless, and use three significant figures in ALL calculations
for this question.
a. What is the tension in hair 1 and in hair 2?
b. What is Homer’s weight in lbs? Note that 1 kg = 2.2 lbs, and use
g = 10 m/s2 for this problem.
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2)
Calvin and Hobbes are in a wagon traveling down a
constant velocity. Assume that the wagon’s total
Calvin and his stuffed tiger Hobbes) is mw . The
an angle θ with respect to the horizontal. Please
“downhill” direction to be to the right.
hill
mass
hill
take
at a
(including
is sloped at
the
a. Derive an expression for the frictional force acting on their
wagon?
b. Derive an expression for the normal force acting on their wagon?
c. Derive an expression for the coefficient of kinetic friction
between the wagon wheels and the hill.
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Honors Physics In-Class Examples
When two objects of unequal masses are hung vertically over a frictionless pulley of negligible mass, we call
this system an “Atwood Machine.” Given the picture, determine the acceleration of the two objects and the
tension in the lightweight cord by deriving equations for each.
a. Assume that m1 = 1kg and m2 = 2kg. What is the acceleration of the system?
b. Assume that m1 = 1kg and m2 = 5kg. What is the acceleration of the system?
c. Assume that m1 = 1kg and m2 = 10kg. What is the acceleration of the system?
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A ball of mass m1 and a block of mass m2 are attached by a lightweight cord that passes over a frictionless
pulley of negligible mass (see diagram below). The block rests on a frictionless incline of angle theta. Derive
expressions for the magnitude of the acceleration of the two objects and the tension in the cord.
For all parts below, assume the angle is 20 degrees.
a. If m1 = 5kg and m2 = 3kg, what is the acceleration of the system?
b. If m1 = 5kg and m2 = 3kg, what is the tension in the cord?
c. If m1 = 5kg and m2 = 1kg, what is the acceleration of the system?
d. If m1 = 5kg and m2 = 1kg, what is the tension in the cord?
e. If m1 = 5kg and m2 = 10kg, what is the acceleration of the system?
f.
If m1 = 5kg and m2 = 10kg, what is the tension in the cord?
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In-Class Practice Problems
EXAMPLE 1 (4.19): Two blocks are fastened to the ceiling of an
elevator, as in figure P4.19 on page 110. The elevator accelerates
upward at 2.00 m/s2. Find the tension in each rope.
EXAMPLE 2 (4.23): A 5.0kg bucket of water is raised from a well by a
rope. If the upward acceleration of the bucket is 3.0 m/s2, find the
force exerted by the rope on the bucket.
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EXAMPLE 3 (4.25): A 2000-kg car is slowed down uniformly from 20.0
m/s to 5.00 m/s in 4.00 seconds. a) What average force acted on the
car during that time, and b) how far did the car travel during that
time?
EXAMPLE 4 (4.28): An object of mass 2.0 kg starts from rest and
slides down an inclined plane 80.0 cm long in 0.50 seconds. What net
force is acting on the object along the incline?
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EXAMPLE 5 (4.32): a) An elevator of mass m moving upward has two
forces acting on it: the upward force of tension in the cable and
the downward force due to gravity. When the elevator is accelerating
upward, what is greater, the tension or the weight force? b) When
the elevator is moving with constant velocity upward, which is
greater, the tension or the weight force? c) When the elevator is
moving upward, but the acceleration is downward, which is greater,
the tension or the weight force? d) Let the elevator have a mass of
1500 kg and an upward acceleration of 2.5 m/s2. Find the tension
force. Is your answer consistent with your answer in part A? e) The
elevator in part d now moves with a constant upward velocity of 10
m/s. Find T. Is your answer consistent with part b? f) Having
initially moved upward with constant velocity, the elevator begins
to accelerate downward at 1.50 m/s2. Find the tension force. Is your
answer consistent with part c?
EXAMPLE 6 (4.45): Objects with masses of m1 = 10.0 kg
kg are connected by a light string that passes over a
pulley as in picture P4.30 on page 111. If, when the
from rest, m2 falls 1.00 m in 1.20 seconds, determine
of kinetic friction between m1 and the table.
and m2 = 5.00
frictionless
system starts
the coefficient
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EXAMPLE 7 (4.46): A car is traveling at 50.0 km/h on a flat highway.
A) If the coefficient of kinetic friction between road and tires on
a rainy day is 0.100, what is the minimum distance in which the car
will stop? B) What is the stopping distance when the surface is dry
and the coefficient of kinetic friction is 0.600?
EXAMPLE 8 (4.48): Objects of masses m1 = 4.00 kg and m2 = 9.00 kg are
connected by a light string that passes over a frictionless pulley
as in P4.48 on page 113. The object m1 is held at rest on the floor,
and m2 rests on a fixed incline of θ = 40 degrees. The objects are
released from rest, and m2 slides 1.00 m down the incline in 4.00
seconds. Determine a) the acceleration of each object, b) the
tension in the string, and c) the coefficient of kinetic friction
between m2 and the incline.
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EXAMPLE 9 (4.57): A boy coasts down a hill on a sled, reaching a
level surface at the bottom with a speed of 7.0 m/s. If the
coefficient of friction between the sled’s runners and the snow is
0.0500 and the boy and the sled together weigh 600.0 N, how far does
the sled travel on the level surface before coming to rest?
EXAMPLE 10 (4.59): A box rests on the back of a truck. The
coefficient of static friction between the box and the bed of the
truck is 0.300. a) When the truck accelerates forward, what force
accelerates the box? b) Find the maximum acceleration the truck can
have before the box slides.
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EXAMPLE 11 (4.65): Two boxes of fruit on a frictionless horizontal
surface are connected by a light string as in Figure P4.65 on page
115, where m1 = 10.0 kg and m2 = 20.0 kg. A force of 50.0 N is
applied to the 20.0 kg box. a) Determine the acceleration of each
box and the tension in the string. b) Repeat the problem for the
case where the coefficient of kinetic friction between each box and
the surface is 0.10.
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EXAMPLE 12 (4.69): Three blocks of masses 10.0 kg, 5.00 kg, and 3.00
kg are connected by light strings that pass over frictionless
pulleys as shown in Figure P4.69 on page 115. The acceleration of
the 5.00 kg block is 2.00 m/s2 to the left, and the surfaces are
rough. Find a) the tension in each string, and b) The coefficient of
kinetic friction between blocks and surfaces. Assume the same
coefficient for kinetic friction applies for all blocks.
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Discussion & Challenge
Questions
Relevant Topics
•
•
Forces and Free Body Diagrams (FBDs)
1. The Free Body Diagram
2. Force Inventory
Friction
1. Kinetic Friction
2. Static Friction
Key concepts
•
•
Free Body Diagrams
Force Inventory
◦
◦
◦
◦
•
Normal Forces (no formula)
Tension Forces (no formula)
Spring Forces
( F  kx )
Gravitational Forces
 W  mg (near Earth surface)
mm
 FGravity  G 1 2 2 (in general)
r
Friction
◦
◦
Kinetic Friction ( f k  k N )
Static Friction
 Has a maximum possible value of s N
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EQUATIONS
Kinematics
g  9.81 sm2  32.2 sft2
v  v0  at
x  x0  v0t  12 at 2
v 2  v02  2a  x  x0 
v A, B  v A,C  vC , B
v A, B  v A,C  vC , B
Uniform Circular Motion
a
v2
r
  2r
v  r
Dynamics
Fnet  ma 
dp
dt
FA, B   FB , A
(near Earth’s surface)
mm
Fgravity = G
(in general)
F = mg
1
2
r2
(where G = 6.67 x 10-11)
Fspring = -kx
Friction
f = kN (kinetic)
f  sN (static)
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Exit Ramp
On a trip to the Colorado Rockies, you notice that when the freeway
goes steeply down a hill, there are emergency exits every few miles.
These emergency exits are straight ramps which leave the freeway and
are sloped uphill.
They are designed to stop runaway trucks and
cars that lose their brakes on downhill stretches of the freeway
even if the road is covered with ice. You are curious, so you stop
at the next emergency exit to take some measurements. You determine
that the exit rises at an angle of 10.0o from the horizontal and is
100.0m long. What is the maximum speed of a truck that you are sure
will be stopped by this road, even if the frictional force of the
road surface is negligible?
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Hanging Sculpture
You are part of a team to help design the atrium of a new building.
Your boss, the manager of the project, wants to suspend a 10.0-kg
sculpture high over the room by hanging it from the ceiling using
thin, clear fishing line (string) so that it will be difficult to
see how the sculpture is held up.
The only place to fasten the
fishing line is to a wooden beam which runs around the edge of the
room at the ceiling.
The fishing line that she wants to use is
known to be able to support a vertically hanging mass of 10.0 kg, so
she suggests attaching two lines to the sculpture to be safe. Each
line would come from the opposite side of the ceiling to attach to
the hanging sculpture.
Her initial design has one line making an
angle of 20.0o with the ceiling and the other line making an angle of
40.0o with the ceiling.
She knows you are taking physics, so she
asks you if her design can work.
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Friction Intro
(a) A block of mass M = 5.0 kg slides on a horizontal table. The
kinetic coefficient of friction between the block and the table is
µ = 0.38. If the initial speed of the block is 8.0 m/s, how many
seconds does it slide before stopping?
(b) A block of mass M = 15 kg is pulled across a horizontal table by
a string. The kinetic coefficient of friction between the block
and the table is µ = 0.69. If the speed of the block is constant
at 2.0 m/s, what must the tension in the string be?
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Friction by Time
A block of mass M is released from rest on an incline of length L
which makes an angle θ with the horizontal. The block reaches the
end of the incline in time T. Show how the coefficient of kinetic
friction can be determined from the measurement of time T.
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Emergency Stop
Your friend has been hired to design the interior of a special
executive express elevator for a new office building. This elevator
has all the latest safety features and will stop with an
acceleration of g/3 in the case of an emergency.
The management
would like a decorative lamp hanging from the unusually high ceiling
of the elevator. He designs a lamp which has three sections which
hang one directly below the other. Each section is attached to the
previous one by a single thin wire, which also carries the electric
current. The lamp is also attached to the ceiling by a single wire.
Each section of the lamp weighs 7.0 N. Because the idea is to make
each section appear that it is floating on air without support, he
wants to use the thinnest wire possible. Unfortunately the thinner
the wire, the weaker it is.
Since he knows that you have taken a
course in physics, he asks you to calculate the force on each wire
in case of an emergency stop.
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Name: ___________________________________ Deriving Equations: Pendulum Period
Possible Quantities that might contribute to your expression: Ɵ, L, t, m, g, π
Collect data in a neat table format. In your groups, decide which data you’ll need to collect. I strongly
recommend varying different quantities to see if they factor into the equation!!!
“Play around” with the data and see if you can come up with an expression for T, which represents the period
(in seconds) for a pendulum. As a hint, you might need a square or a square root somewhere in your
expression!
Your best guess at the equation: ________________________________________________