Physics/Unit 02/BFPM
Balanced Force Particle Model
A force is ____________________________________
Common Types of Forces
Balanced Force Particle Model – BFPM
C BFPM.1
C BFPM.2
C BFPM.3
A BFPM.4
A BFPM.5
C BFPM.6
C BFPM.7
I can draw a properly labeled force diagram showing all forces acting on an
object.
- Identify surrounding objects that interact with an object and the forces
they exert on the object.
- Force vectors are qualitatively accurate (based on direction and size)
I can apply Newton’s 1st Law by relating the balanced/unbalanced forces on an
object to its constant/changing motion and can develop balanced force
equations describing an object with a constant velocity.
- When forces are balanced, the net force must be zero.
I understand and can apply the relationship between mass and weight.
- The gravitational field strength on the Earth’s surface, g, is equal to 9.81
N/kg. (10 N/kg is allowed for basic problems).
I understand and can apply the relationship between friction force and the
normal force on an object.
- The coefficient of friction, µ, is a constant based on the surface of the two
interacting objects.
I can solve balanced force problems using a shifted coordinate axis. (i.e. ramp
problem)
I can demonstrate understanding of Newton’s 3rd Law by identifying force pairs
in multiple situations.
- A force is one half of the interaction between two objects.
I can solve balanced force problems on a horizontal coordinate axis.
-1-
Physics/Unit 02/BFPM
Notes: System Schemas, Free Body Diagrams
-2-
from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Forces and Force Diagrams
Forces can intuitively be thought of as pushes and pulls. For example, you exert a force (a push
or a pull) on a door to open it. Gravity exerts a force on you (a pull) which holds you to the
surface of the earth. Friction with the surface of a hill exerts a force on your car that keeps it from
sliding when parked. Note that in every situation, forces are an interaction between two objects-you can't touch without being touched. The door also pushes back on your hand, the earth is also
gravitationally attracted to you, and the car exerts a frictional force on the road.
There are many types of forces between objects that are differentiated by the way in which two
objects interact. Here are some of the ones we will use in class:
•
•
•
•
When two surfaces touch each other, forces perpendicular to the surfaces are called
normal forces (here "normal" is a mathematical term meaning perpendicular) and forces
parallel to the surfaces in contact are frictional. The Friction force that allows us to step
forward or keeps car wheels from spinning is sometimes called traction. When we touch
things a combination of both normal and frictional forces are present.
Extended or linked materials such as a string or chain exert tension forces on an object.
When an object interacts with a fluid, such as water or air, propelling forces are called
thrust, resistive forces are called drag, floating forces are called buoyant, and steering
(or Bernoulli's) forces are called lift.
When two objects interact without touching, they exert forces through a force field. Earth,
for example, exerts a gravitational force on the Moon even though the Earth and Moon
do not touch. Other non-contact forces include electric and magnetic forces. Technically,
there are NO contact forces at an atomic level. All "contact" forces are actually
gravitational and electromagnetic forces. (why is this true?)
When we label forces, we want to indicate the type of interaction between the objects and what
object is exerting or causing the force. Therefore, we will use the following notation:
FType (partner object)
For example, the gravitational force on you would be written:
Fg (earth)
The analysis of a problem in dynamics usually involves the selection and analysis of the relevant
forces acting on some object under consideration. An important first step in this analysis process
is to carefully select the object of interest that will be the focus of our analysis. For purposes of
this analysis, we will refer to the object under consideration as the system, and everything else
in the environment that might in any significant way affect the system as the surroundings. This
analysis process can often times be greatly simplified by utilizing a technique of constructing
force diagrams to assist you in selecting the relevant forces and appropriately representing
these forces with vector notations.
In
1.
2.
3.
4.
5.
general, we will follow the following steps when creating force diagrams.
Sketch the system and its surroundings with a System Schema
Enclose the system within a system boundary.
Shrink the system to a point because we will represent the system as a particle.
Represent all relevant forces (across the system boundary) with a labeled force vector.
Indicate which forces (if any) are equal in magnitude to other forces with congruent lines.
-3from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Consider the analysis of forces acting on a log as a tractor pulls it at a constant speed. The
analysis proceeds as follows:
Step 1. Sketch a diagram of the
system and its surroundings.
Log
Rope
Ground
Earth
Step 2. In order to assist in the identification of the
relevant forces acting on the system, enclose the
system (log) within a closed boundary line.
Tractor
Log
Rope
Ground
Earth
Tractor
Step 3. Since the shape of the object is unimportant, shrink it to a
point or a box. Place it at the intersection of a set of coordinate
axes with one of the axes parallel to the direction of motion as
shown in figure 4.
Step 4. Proceed around the system boundary line and identify all
points at which there is contact between the system (log) and its
surroundings. Construct qualitative vectors (indicate directions and
relative magnitudes) to represent these forces. In this example,
there is one force in each direction. That is not always true.
FN (ground)
Ff (ground)
FT (rope)
Log
Fg (Earth)
Step 5: Indicate which forces (if any) are equal in magnitude
to other forces. The problem states that the tractor pulls the
log at constant velocity, so we know that the net force has to
be zero. In other words, the forces up must equal the forces
down, and the forces left must equal the forces right. In the
diagram below these equalities have been marked with hashes
like those used to indicate congruencies in geometry.
FN (ground)
Ff (ground)
Log
FT (rope)
Fg (Earth)
-4from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Practice 1: Forces and Motion (BFPM.1)
Take care in reading every word in these questions. Make sure you know exactly when
we are taking our snapshots in each part of the problem. When making a whiteboard,
arrange your work so that all parts of one question can fit on one board (neatly).
1. A heavy cardboard box has a rubber bottom and is sitting at rest on a rough, concrete,
horizontal floor.
a. Draw a system schema for this
situation.
b. Create a force diagram. Show and clearly label each
force on the system. Make it obvious from your
diagram which forces you intend to be equal and
which you intend to be greater than others.
c. A person shoves the box horizontally so that it begins to move. Your answers to this part
should concern the time while the person is still touching the box and shoving. (i) Draw a
velocity-vs-time graph for the box, clearly marking the time when the box is at rest and the
time when the person is still touching the box and shoving it. This should be qualitatively
accurate (no numbers, but correct shape). (ii) Also draw a system schema for that same
time period. (iii) Also draw a force diagram for the box during that same time period. Label!
d. The shove ends when the box leaves contact with the person’s hands. (i) Draw a
qualitatively correct velocity-vs-time graph for the box, clearly marking the time when the
box is at rest, the time when the person is still touching the box and shoving it, and the
time after the box loses contact with the person’s hands. Make it obvious which lines are
horizontal, which have greater slopes, smaller slopes, or negative slopes.
(ii) Also draw a system schema for the box when it leaves contact with the person’s hands.
(iii) Also draw an Force Diagram.
-5from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
2. The rubber is now removed from the bottom of the box so that the cardboard surface
rests directly on the same floor. It is then given a horizontal shove by a person. In the
space below, modify your velocity vs. time graph as well as your system schemas and force
diagrams from problem 1 to accurately describe this new situation. Your diagrams do not
have to be quantitatively accurate, but make it obvious which forces you intend to be equal
and which you intend to be greater or less than others, so that comparisons can be made
among forces in this problem as well as between forces in this problem and in problem 1.
Make any differences in your graphs and diagrams obvious.
a. Draw a velocity-vs-time graph for the box, clearly marking the three time periods.
b. Draw a system schema and a force diagram for this situation while the box is at rest on
the horizontal floor.
c. Draw a system schema and a force diagram for this situation while the person is still
touching the box and shoving.
d. Draw a system schema and a force diagram for this situation during the time after the
box loses contact with the person's hands.
-6from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
3. Suppose that we could somehow succeed in making the floor completely frictionless.
Again, make new diagrams/graphs to represent this new variation in the situation. Make
any differences obvious.
a. Draw a velocity-vs-time graph for the box, clearly marking the three time periods.
b. Draw a system schema and a force diagram for this situation while the box is at rest on
the horizontal floor.
c. Draw a system schema and a force diagram for this situation while the person is still
touching the box and shoving.
d. Draw a system schema and a force diagram for this situation during the time after the
box loses contact with the person's hands.
-7from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
5. Return to the case of the cardboard box resting on the concrete floor, with friction. A
person pushes it hard enough to start it into motion and then continues pushing so that it
maintains a constant velocity. In the space below, modify your velocity vs. time graph as
well as your system schemas and force diagrams from problem 2 to accurately describe this
new situation.
a. Draw a velocity-vs-time graph for the box, clearly marking the three time periods (at
rest, velocity is changing, moving with a constant velocity).
b. Draw a system schema and a force diagram for this situation while the box is at rest on
the horizontal floor.
c. Draw a system schema and a force diagram for this situation while the box is changing
velocity.
d. Draw a system schema and a force diagram for this situation while the box moves with a
constant velocity.
e. After pushing the box with a constant velocity for a while, you reduce your force to half
the value needed to maintain a constant velocity. Make a new (continued) velocity-vs-time
graph to show what happens to the box while you continue to push with this force
-8from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Bowling Ball Motion
As part of our study of motion and forces you will push a bowling ball across the floor with
the bristles of a broom. For each of these situations, describe (using words, pictures,
system schema, force diagrams) how to accomplish each feat.
Make the bowling ball speed up from rest.
Stop a moving bowling ball.
Keep a moving bowling ball moving at constant velocity.
Move the ball from one line to another and back as quickly as possible,
without overshooting either line.
-9from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
With a moving bowling ball, make a sharp left turn.
Travel at constant speed along a curved line.
Move the ball around a circle as quickly as possible.
-10from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Practice 2: Force Diagrams (BFPM.1)
In each of the following situations, create a system schema and represent the
object with a labeled force diagram. Label each force with a meaningful symbol
(ex. Fg) AND with the object exerting the force (ex. Fg(earth)).
-11from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
-12from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Interactions between Two Objects
For each of the situations below, create a system schema and force diagram.
Situation 1
A water balloon is set on a spring one meter off the ground. Consider
the water balloon and the spring.
In what way does each object influence the other?
Situation 2
A child stands on a wooden plank joining two large boulders on
opposite sides of a brook. Consider the child and the plank.
In what way does each object influence the other?
Situation 3
A block sits at rest on a horizontal table. Consider the block and the
table.
In what way does each object influence the other?
1.
In situation 1, if the spring were to suddenly disintegrate, what would happen
to the water balloon?
2.
(a) In situation 2, is the child interacting (directly) with the boulders?
(b) With what object(s) is(are) the child interacting directly?
3.
(a) In what ways are the spring, the plank, and the table the same?
(b) How are they different?
-13from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Identifying All Possible Interactions
For each of the situations below, (a) create a system schema, (b) create a force
diagram, (c) indicate if the object is changing its motion, and (d) create a motion
map for the object.
Situation 1
The space shuttle moves away from the launch pad
just after take-off.
Situation 2
A book slides across the floor before coming to rest after 2m.
Situation 3
A coffee mug sits on a table with a giant dictionary on top of it.
Thought questions
1. Under what conditions is air an important agent? (That is, under what conditions
must we include interactions of an object with the air surrounding it?)
2. Under what conditions can we ignore the effects of the air surrounding an object?
(For example, can we ignore the effects of air when something is traveling
very fast?)
-14-
from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Investigating Force of Gravity (BFPM.3)
What could affect the situation/what could we measure?
Sketch and label the experimental setup:
The Objective/Lab Question:
Your hypothesis (written and graphical):
Take data in an organized way with proper measurement and uncertainty labels:
-15from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Graph your data to determine the relationship
Additional analysis (i.e. calculating slope, y-intercept, and error analysis).
The mathematical model for your data: ___________________________________
The meaning of your slope and y-intercept.
The General Equation: ______________________________________
-16from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Investigating Force of Springs
What could affect the situation/what could we measure?
Sketch and label the experimental setup:
The Objective/Lab Question:
Your hypothesis (written and graphical):
Take data in an organized way with proper measurement and uncertainty labels:
-17from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Graph your data to determine the relationship
Additional analysis (i.e. calculating slope, y-intercept, and error analysis).
The mathematical model for your data: ___________________________________
The meaning of your slope and y-intercept.
The General Equation: ______________________________________
-18from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Investigating Friction Forces
What could affect the situation/what could we measure?
Sketch and label the experimental setup:
The Objective/Lab Question:
Your hypothesis (written and graphical):
Take data in an organized way with proper measurement and uncertainty labels:
-19from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Graph your data to determine the relationship
Additional analysis (i.e. calculating slope, y-intercept, and error analysis).
The mathematical model for your data: ___________________________________
The meaning of your slope and y-intercept.
The General Equation: ______________________________________
-20from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Reading 2: Scalars, Vectors, and Trigonometry
Some physical quantities like time, mass, and speed are characterized by having only a
size, also referred to as magnitude. Such quantities are called scalars. Other physical
quantities like displacement, velocity, acceleration, and force have both magnitude and
direction. Such quantities are called vectors.
In the previous units, we have been able to indicate the direction of the velocity by
using positive and negative signs because we only considered motion along a line.
Much of our study of forces will be two-dimensional, requiring us to be more specific
when indicating direction. We must also be more careful when adding vectors in two
dimensions. To perform a vector sum, we need to be able to take the vectors that
aren't aligned with a coordinate axis and align them with
the coordinate axes. Then, vectors along each axis can be
added.
To align a vector with a coordinate axis, we represent an
unaligned vector, such as A on the right, with two
component vectors, such as Ax and Ay, whose sum is the
same as the original vector. Each component vector is
aligned along one of the coordinate axes and they are
perpendicular to one another.
Again, since Ax and Ay are not in the same direction, adding their magnitudes will not
give you the value of the magnitude of A. However, this is a right triangle, so you
could use the Pythagorean theorem to relate the magnitudes of the sides.
Everything you need to know about trigonometry for this course:
Once upon a time, someone noted a very interesting property
of similar right triangles. If you take the ratio of any two sides
on the little triangle, such as a/b, the ratio is exactly the same
as A/B for the big triangle. The value of the ratio corresponds
to a specific angle, θ (theta).
For example, if A/B = 1, then θ = 45° no matter how big or small the triangle is. If a/c
= 1/2, then θ = 30° no matter how big or small the triangle is. If someone would sit
down and work out every pair of ratios for every shape of right triangle (which has
been done), then when we knew the angle, we could use the value of the ratio
between two sides to find the length of an unknown side. For example, suppose we
know θ is 30° and A is 10 meters long. Since the ratio of A/C is 1/2 for a 30° right
triangle, we know that C must be 20 meters long.
To formalize what has just been said, let us first name each of the sides relative to
angle θ. The side next to the angle is called the adjacent side, the one across the
triangle from the side is called the opposite side, and the longest side of a right
triangle is called the hypotenuse.
-21from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Our next bit of formalizing involves naming the ratios of the sides. Although we could
name six, we will just use three, since the other three are reciprocals of the first three.
Sine of angle θ is the ratio of the opposite side divided by the
hypotenuse.
Cosine of angle θ is the ratio of the adjacent side divided by the
hypotenuse.
Tangent of angle θ is the ratio of the opposite side divided by the
adjacent side.
Shorthand notation:
So whenever you see sin(45) or tan(21) just remember that these are just numbers;
sin(45) is a decimal number that results from dividing the opposite side by the
hypotenuse in a 45-45-90 triangle. You can get the value of the ratio from your
calculator or from a trigonometric table. If you use your calculator, be sure that it is in
degree mode.
To find the angle when the sides of a triangle are known, the inverse trigonometric
relations are used. In other words, given a ratio, what is the corresponding angle? On
a trigonometric table this is easy to see. If the tangent ratio for a triangle is 0.8, then
we can look in the tangent column until we find a value of 0.8 and then look over to
see that the corresponding angle is 39°. On your calculator, you would hit "inverse
tangent" which is "2nd" "tan" on most calculators and then enter the ratio. The
calculator will then display the angle.
Shorthand notation:
(Note that the -1 is not an exponent meaning "take the reciprocal" in this case.
Instead it means, "find the angle that corresponds to this ratio of triangle sides.")
Example: In order to find the component vectors, Ax and Ay, we can use the
trigonometric relations we have just identified.
Now we could plug in the magnitude for vector A and the angle
for θ. Taking the sine and cosine of θ will give us a decimal in
each case that we can multiply with the magnitude of vector A.
If we already knew the values of Ax and Ay, we could use inverse tangent to find the
angle.
-22from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Component Vector Practice
Determine the x and y components of each of the force vectors below. Show work.
1.
2.
3.
4.
-23from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Practice 3: Interaction Problem Solving
The player in the photo exerts a 100 N horizontal force on a 25 kg blocking sled, pushing it
across the grass with a constant speed of 2.0 m/s.
a. Draw the system schema and force diagram.
b. How would the situation change if the player pushed with more than 100 N, while the
frictional force between the grass and the sled remained the same? Illustrate your answer
with another force diagram and a motion map.
c. Describe, in terms of the amount of force he would have to apply, what the player would
have to do to make the sled move with a constant velocity of 3.0 m/s. Assume that the
frictional force between the grass and the sled remains the same under all circumstances.
Illustrate your answer with diagrams and/or graphs as appropriate.
d. If he pushes the sled as originally described with a velocity of 2.0 m/s, how far will it
slide in 7.5 seconds? Draw at least three diagrams/graphs to illustrate this situation, then
solve this problem using at least two different methods (and getting the same answers).
e. With the sled moving at a constant velocity of 2.0 m/s, the person reduces his force to 75
N. Describe what happens to the sled. Illustrate your answer with another FBD and motion
map.
-24from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Practice 4: BFPM Problem Solving
A person pulls on a 50 kg desk with a 200N force acting at 30°
angle above the horizontal. The desk does not budge.
a. Draw the system schema and force diagram for the desk.
b. Using your force diagram, develop force equations in the horizontal and vertical
directions.
c. Use Trig functions to determine the x and y components of the force of tension.
d. Use your force equations to determine the value of the frictional force and the
normal force.
6. Suppose in the diagram above, the person were pushing down at a 40° angle with
100 N of force. The desk still does not move.
a. Draw the system schema and force diagram for the desk.
b. Using your force diagram, develop force equations in the horizontal and vertical
directions.
c. Use Trig functions to determine the x and y components of the force of tension.
d. Use your force equations to determine the value of the frictional force and the
normal force.
-25from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
A man pulls a 50 kg box at constant speed across the floor. He applies a 250 N force
at an angle of 30°.
a. Draw a force diagram.
b. Use your horizontal force equation to determine the value of the frictional force
opposing the motion?
c. Use your vertical force equation to determine the value of the normal force?
A man pushes a 2.0 kg broom at constant speed across the floor. The broom
handle makes a 50° angle with the floor. He pushes the broom with a 5.0 N force.
a. Draw a force diagram.
b. Use your horizontal force equation to determine the value of the frictional force
opposing the motion?
c. Use your vertical force equation to determine the value of the normal force?
d. If the frictional force were suddenly reduced to zero, what would happen to the
broom?
-26from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Practice 5: Problems with a Shifted Coordinate Axis
For each of the problems below, carefully draw a force diagram of the system before attempting
to solve the problem.
1. The object hung from two cables has a weight of 25 N. Include system schema,
force diagram, force eqns, and trig. What is the tension in each cable?
2. In the system below the pulley and ramp are frictionless and the block is in static
equilibrium. What is the mass of the block on the ramp?
3. The cable at left (T1) exerts a -30 N force. Use the connector point as your
system/particle for the force diagram.
40ο
T2
a. Using your horizontal force equation, determine the value
of T2?
b. Using your vertical force
equation, determine the force
of gravity acting on the ball?
T1
-27from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
4. The box on the frictionless ramp is held at rest by the tension force. The mass of
the box is 20 kg. Include system schema, force diagram, force eqns, and trig in
your analysis.
T
What is the value of the tension force?
What is the value of the normal force?
5. Determine the values of m1 and m2 that balance the forces on the 1 kg mass so that
the mass could stay balanced if you were to remove the ramp. Include system
schema, force diagram, force eqns, and trig in your analysis.
-28from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Dueling Forces
For each of the following situations:
1. Draw one system schema. In your system schema, draw the interaction between the two
carts in colored pencil. (Keep everything else in regular pencil.)
2. Draw and label two FBDs (one for each cart). Draw the forces the carts exert on one
another in colored pencil. (Again, keep everything else in regular pencil.) Be sure your
FBDs look balanced or unbalanced as appropriate. Draw forces to approximate scale.
3. Finally, measure the colored pencil forces with the force sensors and correct your
diagrams if necessary. Remember to zero your force sensors!
4. After completing the ones on this sheet, if you have time (or outside of class), you might
be interested in trying additional variations and confirming your results.
I. You may ignore friction on this particular situation.
II. Do not ignore friction.
-29from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
III. Do not ignore friction.
IV. Do not ignore friction.
V. This should be a collision on a track (snapshot during the collision). You may ignore
friction in this situation.
-30from Modeling Workshop Project © 2006
Physics/Unit 02/BFPM
Horse and Cart Scenario
1. Draw a system schema
This horse is pulling the cart causing it to speed up.
2. Draw a force diagram for the horse and for the cart.
3. Is the force exerted on the cart by the horse greater than the force exerted on the
horse by the cart? How do you know?
4. Is the force exerted on the cart by the horse greater than the force of friction on the
cart exerted by the ground? How do you know?
5. Is it true to say that the net force exerted on the cart is in the forward direction?
How do you know?
6. Is it true to say that the net force exerted on the cart is zero? How do you know?
7. Is it true to say that the net force of each Newton’s third law pair is zero? How do
you know?
8. Is the force of friction on the horse less than the force of friction on the cart? How
do you know?
-31adapted from Hewitt Conceptual Physics
Physics/Unit 02/BFPM
Practice 6: Newton's 3rd Law in Action
A block slides down a ramp at a constant speed. During that slide,
the ramp sits at rest on a table. Draw one system schema for the
situation, then draw a force diagram for the block and a force
diagram for the ramp.
In frustration, Alec gets Henry to hold up his test and punches his fist completely through all of
the sheets of paper. Which is greater: the force that Alec’s fist exerted on the paper or the force
that the paper exerted on Alec’s fist? Explain.
Your friend’s truck stalls out on a hill, so you get out to push. However, after a couple minutes
you start to tire yourself out and the truck starts pushing you back down the hill. While the
truck is pushing you back down the hill, which is greater: the force that you exert on the truck
or the force that the truck exerts on you? Explain.
At the ice skating rink, Lydia (who has a mass of 50 kg) stands face to face
with her brother, Marcus (who has a mass of 80 kg). They put their hands
together and Lydia pushes Marcus backwards. Draw one system schema and
two FBDs (one each for Lydia and Marcus) during the push. You may assume
that the ice is frictionless.
-32from Modeling Workshop Project © 2006