Team ________________
________________
Force and Motion
In previous labs, you used a motion sensor to measure the position, velocity, and acceleration of
moving objects. You were not concerned about the “mechanism” that caused the object to move,
i.e. the forces that acted on the object. In other words, you answered the kinematic question
“How do objects move?” Now it is time to answer the dynamic question “Why do objects
move?” From your experiences, you know that force and motion are related. To open a door,
you must push or pull on the handle. To stop a car, you must step on the brakes. The force of
gravity causes the earth to move around the sun. The force of electricity causes the electron in an
atom to move around the proton. The strong quantum force “glues” the protons (quarks) inside
the tiny nucleus of an atom.
Newton’s celebrated equation of motion, F = ma , describes the precise connection between
force (F) and motion (a) in classical mechanics. You have used a motion sensor (accelerometer)
to measure a. You will now use a “force-ometer” to measure F. By measuring the left side of
F = ma with a force meter and the right side with a motion meter, you will discover the deep
relation between F and ma firsthand in the laboratory.
Thought Experiment
A 2 kg mass and a 3 kg mass are connected by a massless cord and move on a horizontal
frictionless surface. The 3 kg mass is pulled to the right with a force of 20 N. The tension in the
cord is 8 N. Fill in the “F” and “ma” blanks.
2 kg
8N
20 N
3 kg
Force:
Net Force on 2 kg mass
=
__________ N .
Net Force on 3 kg mass
=
_____
−
_____
=
__________
N.
Motion:
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Mass x Acceleration of 2 kg mass =
_____ ×
_____
=
__________
kg m/s .
Mass x Acceleration of 3 kg mass =
_____ ×
_____
=
__________
kg m/s .
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Part I. “Forceometers”
The Spring Scale
The spring scale is the prototype “forceometer”. For centuries, springs have been used to
measure force. The force required to stretch a spring is proportional to the amount of stretch. A
spring scale is a spring that has been calibrated to convert the amount of stretch (meters) into the
corresponding value of the force (Newtons).
Examine the spring scale. Note that there is a large fixed hook at one end and a small hook at the
other end. The small hook is attached to a sliding rod, which is connected to the spring. Pull on
the small hook so that the spring is stretched several centimeters (half its maximum stretch) and
the reading on the force scale is 0.5 N. Double the stretch of the spring so that the force is 1.0 N
(maximum scale reading).
You already know what a “one pound force” feels like. But what does a “one Newton force” feel
like? Pull on the spring scale until you have a good kinesthetic sense of a one Newton force.
Zero the spring scale as follows. Hold the spring scale in the vertical position with the small
hook downward. With nothing attached to the hook, make sure the scale reads zero. If the
reading is not “0” , then turn the screw at the top of the tube until zeroed. Once the scale is
zeroed, hang a 50 gram mass from the small hook. What is the value of the force F recorded by
the spring scale?
Spring Scale Reading:
F =
______________ N.
Since the 50 gram mass is at rest, the upward pull of the spring on the mass is equal to the
downward pull of the earth on the mass. The pull of the earth − the force of gravity − on an
object is called the Weight of the object. Calculate the theoretical value of the weight of a
50 gram mass.
Calculated Weight of 50 gram mass:
W = ______________ N.
% diff between Measured Weight (spring scale reading) and Calculated Weight is ______ %.
The Force Sensor
Examine the force sensor. The switch on the force sensor should be set to the 10 N range. The
force sensor is an electronic strain gauge. Any force (push or pull) on the hook causes a metal
block inside the sensor to bend slightly. The strain in the block is converted into an electrical
signal (voltage) which is then converted, via the interface box, into the corresponding value of
the mechanical force. So like the spring scale, the force sensor converts a distance (strain in
meters) into a force (Newtons).
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The force sensor measures F(t) = Force on the hook as a function of time. To activate the force
sensor or “force probe”, start Logger Pro and open the file ForceProbe. Click on Collect. Push
and pull on the hook and note the digital values of F(t) that are displayed in the data table and the
corresponding F(t) curve that is traced out in the graph window.
Zero the force sensor as follows. While holding the force sensor in the vertical position with the
hook downward and nothing attached to the hook, click on the Zero button (to the right of the
Collect button). Once the sensor is zeroed, hang a 50 gram mass from the hook and click on
Collect. Average the F(t) data. What is the value of the force F recorded by the force sensor ?
Force Sensor Reading: F =
% diff between
______________ N.
Measured Weight (force sensor reading) and Calculated Weight is ______ %.
Part II. Apply F. Measure ma. Prove the Law F = ma .
Measure F with the Spring Scale
First make sure your track is level using the Steel Ball Test: Place a steel ball at different
locations on your track. If the ball remains at rest, then the table is level. If the ball rolls, then
the track is not level. Adjust the tilt by delicately turning the leveling screw underneath one end
of the track.
The spring scale will be used to apply the force F. Make sure that the spring scale reads “zero”
when it is held in the horizontal position and nothing is attached to the small hook. Attach the
small hook of the spring scale to the cart. Pull the cart along the track with a constant force of
0.10 N. Keep the spring scale tube horizontal (do not tilt the tube). It may be a bit tricky to keep
the force constant. In experimental physics, nothing is exactly constant. It is natural for the
force to fluctuate around 0.10 N. This error in keeping the force constant should have little affect
on your results. Sometimes you pull too hard, sometimes you pull too soft, but on average you
pull just right!
Practice pulling the cart until you get a “feel” for how to keep the force scale reading at 0.10 N.
While you are pulling, carefully note the amount by which your applied force F varies above and
below 0.10 N. This up and down variation in F determines the uncertainty in F. For example, if
the spring scale reading varies between 0.070 N and 0.13 N during the pull, then you would
report the experimental value of the force as F = 0.10 N ± 0.03 N. If your pull is steadier, then
the uncertainty in the force will be smaller.
Based on your pulling trials and observed range of force values, report the measured value of
your applied force:
F =
0.10 N
±
________ N .
Significant Figures: The measurement of force (with the spring scale) and the measurement of
time (with the stopwatch) are accurate to about two, perhaps three, significant figures. All final
results that you report in this lab should reflect this “2-3 sig-fig” accuracy.
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Measure ma with a Stopwatch
An accurate way to determine the constant acceleration a of your cart is to measure time and
space. Use a stopwatch to measure the time t it takes for your cart − starting from rest and
pulled with the force 0.10 N − to cover the distance d = 0.80 m along the track. Use the
kinematic relation d = ½ at2 to compute a from your measured values of t and d. Repeat the run
five times. Record your five measured values of t in the motion table below. For each run,
compute the values of a and ma .
Measure the mass m of your cart using the mass scale.
Motion Table
d (m)
0.80
0.80
0.80
0.80
0.80
t (s)
a (m/s2)
m = _____________ kg .
ma (N)
Compute the average value of ma. Estimate the uncertainty from the half-width spread in your
five values of ma : Uncertainty = (mamax − mamin) / 2.
ma = _____________
±
____________ N .
Compare ma and F
You have measured the force quantity F using a spring scale. In a completely independent
measurement, you have measured the motion quantity ma using a stopwatch (and meter stick and
mass scale). Here is the BIG question: Is F equal to ma ?
Remember, since an experimental value (number ± uncertainty) is really a range of numbers,
“comparing two values” really means “comparing two ranges”. Display your results for F and
ma on the following range diagram. Remember that an experimental value, such as 0.10 ± 0.03
N, is displayed on the range diagram as a horizontal line segment from 0.07 N to 0.13 N. Write
the numerical values on the axis labeled “Newtons”.
F
ma
Newtons
Use the results displayed on your experimental range diagram to rigorously answer the deep
question: Is F equal to ma ?
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What Happens to a if you Double F or Double m ?
Here you will ponder this Newtonian question, first in theory (analyze symbols) and then in
experiment (measure numbers).
Theoretical Prediction
Let a(F) denote the acceleration of a mass due to the net force F.
Let a(2F) denote the acceleration of the same mass due to the net force 2F.
Then
a (2 F )
a(F )
=
__________
.
Let a(m) denote the acceleration of mass m due to a net force.
Let a(2m) denote the acceleration of mass 2m due to the same net force.
Then
a (2m )
a (m )
=
__________
.
Experimental Test of Prediction
Do a short experiment to test your theoretical prediction for the a(2m)/a(m) ratio.
Note: the mass of the long rectangular bar on your table is equal to the mass of your cart.
This “ten-minute experiment” does not have to be high precision (20% is okay) but it should be
convincing. In the space below, briefly describe your experiment and results.
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Part III. Design Problem: Pulling Coupled Train Cars
In the previous part, you studied a system consisting of one mass (body). Here you will study a mechanical
system composed of two connected masses. In physics, the “two-body problem” refers to any dynamical
problem involving two interacting objects. Solving the equations of motion for the sun-earth-moon system
is the famous “three-body problem” in theoretical astrophysics.
You will also discover how the net force − the sum of all the individual forces − acting on a body is the
crucial dynamical quantity that determines the motion of the body. MAY THE NET FORCE BE WITH
YOU!
Design Goal: Pull two coupled train cars (mass m1 and mass m2 initially at rest) with a constant
force Fpull over a given distance d in the shortest possible time t .
m1
F12
Fpull
m2
System Constraint: The coupling force F12 between car 1 and car 2 cannot exceed some critical
value F12max . If F12 > F12max , then the mechanism (“rope”) holding car 1
and car 2 together will break.
In terms of the system parameters, here is the design problem:
Given: m1 , m2 , d , and F12max .
Find:
Fpull and t .
The Theory
1. Draw two free body diagrams, one for body 1 and one for body 2.
2. Set up Fnet = ma for m1 and m2. Symbols only. No numbers.
Solve the m1 equation for t as a function of m1 , d , and F12max . Remember d = ½ at2 .
Solve the m2 equation for Fpull as a function of m2 , d , F12max , and t .
t
Fpull =
=
_______________ .
________________________ .
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The Experiment
Build the following small scale model of the coupled-train system. Body 1 consists of cart 1 plus
long bar weight plus spring scale 1. Body 2 consists of cart 2 plus spring scale 2. Make sure
each spring scale reads “0” when held in the horizontal position and nothing is attached to the
small hook. Note the orientation of the scales: large hook on left, small hook on right. Use two
rubber bands (widely spaced) to secure each scale to its cart. Make sure that the rubber bands do
not rub against the wheels underneath the cart.
scale 1
scale 2
Cart 1
Cart 2
Bar Weight
Experimental Goal: Pull this system (initially at rest) over a distance d = 1.0 meters in the
shortest possible time t given that F12 cannot exceed 0.20 Newtons.
Use your theoretical results derived above to calculate Fpull and t.
Fpull (theory) =
t (theory) =
____________ N .
____________ s .
Perform the experiment. Place your two-cart system directly on the table surface, not on the
metal track. This will allow you to pull the carts over a longer distance. Hold the small hook of
spring scale 2 and pull so that the scale reads your predicted (theory) value of Fpull . As you pull
over the one-meter distance, measure t (use the stopwatch) and F12 (monitor the spring scale).
Record these two experimental values:
t (exp) =
___________ s .
Percent difference between t (theory) and t (exp) is
F12 (exp) =
___________ N .
_______ %.
Percent difference between F12 (given) = 0.20 N and F12 (exp) is
_______ %.
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Part IV. Creating a Physics Sculpture
Title:
“Motionless Bodies on a Tilted Beam.
Gravity and Tension in Perfect Harmony”
Cart
Weight (150 gram)
Track
Box
Depending on the tilt of the track, the cart will accelerate up or down. However, there exists one
special tilt angle such that the two-mass system is in “Equilibrium” − a special state where the
masses do not move at all! In this state of zero acceleration , the net force on each mass is
equal to zero. In the sculpture above, the forces of gravity and tension are in perfect balance.
Design Goal: Where should you place the box so that the masses are motionless ? Work out
the theory behind this mechanical structure before you actually build the structure.
The Theory
Here is a schematic of the system showing the system parameters m1 , m2 , L , H .
m1
m2
L
H
θ
Answer the following questions using only symbols. No numbers allowed.
1. Draw two free body diagrams, one for m1 and one for m2 .
m1
m2
2. Set up Newton’s equations: Fnet 1 = m1 a for body 1 and Fnet 2 = m2 a for body 2.
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3. Solve Newton’s equations to find L as a function of m1 , m2 , and H. Hint: Express sinθ in
terms of L and H.
L =
______________
.
Measure the values of H and m1 . The value of m2 is given to be 150 grams.
m1 = _________ kg .
H = _________ m .
m2 =
0.150
kg .
Plug the values of these system parameters into your theoretical formula for L.
L (theory)
=
_______________
m.
Note that L is the key architectural element in the design because it is the system variable that
determines the overall geometry (tilt) of the sculpture.
Building the Sculpture
Use the theoretical design specs − your calculated value of L − to build the actual sculpture.
Construction Hints: Remember to measure L from the point where the track touches the table to
the point where the track touches the box. Since you assumed tension T was parallel to the track
in your theory, you must make sure the string is parallel to the track in your sculpture. Since
you neglected friction in your theory, remove the end bracket on which the string rubs.
Did your team achieve “perfect balance” the moment you placed the box at L (theory) ? Explain.
Compare L (actual) with L (theory).
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