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How to Design an Experiment and Most Common AP Physics 1 Labs
The following steps outline a general procedure for designing an experiment.
a) Make a MATHEMATICAL MODEL of the experimental situation that relates what you want to determine
(Z), to other variables. The model is obtained by applying physics to the experimental situation to derive an
expected equation or model that contains the unknown variable, Z. That model or equation guides your
experimental design; in other words it tells you what needs to be measured to find Z. [Note that in order to
model the experimental situation, simplifying assumptions usually need to be made (such as neglecting air
resistance or friction) and these assumptions are a source of experimental error).]
Example1: you want to design an
experiment to determine the kinetic
coefficient of friction,k, between a
block and the table. An equation
that contains k is given by
fk = kFN.
This is a model.
Example2: You need to determine the acceleration due to gravity, g, given
a car on a ramp, a meterstick and stopwatch. To design the experiment, you
need to make a model of the car on the ramp using physics. By applying
Newton’s 2nd Law to the car traveling down the ramp, one gets the equation:
acar = gsin
This is a model of the experimental situation. It involves an assumption –
that the force of friction between the car and ramp is negligible. (Note:
Another model could be obtained using conservation of energy: gh = ½ vB2)
b) Use the model to SELECT TWO VARIABLES TO MEASURE:
Example1: Given the model
fk = kFN.
you could experimentally determine k by
measuring the 2 variables:
fk, the force of friction and
FN, the normal force on the block
Example2: Given the model of the experimental situation,
acar = gsin
you could experimentally determine g by measuring the 2
variables:
acar, the acceleration of the car and
, the angle of the incline.
c) EXPERIMENTAL DESIGN: To design the experiment, decide which measured variable is independent
and which is dependent. If the variable cannot be measured directly with accuracy, measure something else
that can be used to determine that variable. For each variable, describe what and how it’s measured. Control
and document other variables. If a measurement has a lot of uncertainty (variability), repeat and average a
few identical trials. Measure at least 6 independent data points (6 different pairs of indep, dep variables).
Example1: Indep var: FN
Dep var: fk
Procedure (what and how):
FN: FN = mg if pull block along horizontal surface.
fk: cant measure directly, but if pull block with
spring scale at constant speed then Fpull = fk.
Constants and controls – same block and table used
throughout, same surface area of block in contact
with table throughout.
1. Measure weight of block (=FN). Using a spring
scale, drag block at a constant speed along the
table so that the reading on the spring scale is
equal to fk.
2. Add mass on top of the block and repeat step 1 to
measure fk for at least 6 different values of FN.
Example2: Indep.Var:
Dep.Var: acar
Procedure (what and how):
: Cant measure with a meterstick, but can measure h and
d along incline and use those to determine (sin = h/d).
acar: cant measure directly with meterstick or stopwatch,
but can measure distance car travels along incline, x,
(starting from rest) and the time it takes, t (with a
stopwatch), and use those to determine acar (x= ½at2).
Constants and controls – same car and ramp used
throughout (car and ramp have minimal friction)
1. Car is released from rest at top of incline. Acceleration
of car, acar is determined for at least 6 different ramp
angles, (as described above).
d) ANALYSIS - Plot and fit the dataset: Use your model to guide the analysis of the data. Make a graph of
the data (with dependent variable on y axis and independent variable on x-axis). Using your model which
gives you the expected relationship between the variables you measured, fit the data to an appropriate
function. Compare the bestfit equation (experimental results) to the model in order to determine the
unknown quantity or to support what you expected.
Example1:
Model or expected: fk = kFN.
Indep var: FN
Dep var: fk
Plot fk vs FN (y vs x)
Fit with a line (because expected relationship is linear)
Compare experimental bestfit to expected model:
slope of bestfit equation (FN/fk) represents uk
y-intercept of bestfit represents error.
Example2:
Model or expected: acar = gsin
Indep var:
Dep var: acar
Plot acar vs sin (y vs x)
Fit with a line (because the expected relationship is linear)
Compare experimental bestfit to expected model:
slope of bestfit equation (a/sin) represents g
y-intercept of bestfit represents systematic error due to
the unaccounted for friction between the car and track
(expect that when =0, a = 0. If a=0 at some nonzero ,
then there is an unaccounted for force in the model and
that was friction)
e) ERROR ANALYSIS: Find %error of experimentally determined value by comparing it with an accepted
value. Cite and discuss sources of systematic errors that would cause the %error. Systematic errors come
from assumptions that were made in applying the model to the experimental situation and from systematic
experimental errors.
Example1: Compare the experimentally
determined k to an accepted value if available to
determine %error. Most likely systematic errors
that caused the %error:
- Notable assumption made in this experiment was
that Fpull = fk. However, if the object is not
moving at constant speed, then Fpull > fk and that
leads to overestimate of fk and k
- Possible experimental systematic errors – Fpull not
parallel to surface and so not just balancing fk
Example2:
Compare the experimentally determined g to 9.81m/s2 to
determine %error. Most likely systematic errors that caused
the %error:
- Notable assumption made in this experiment was that there
is no friction between the car and the track. In reality, the
friction would cause a=0 at a small angle. Unaccounted for
friction would lead to an underestimate in g because the
slope that we assumed to be g based on the model would be
<g if there were friction (a and were measured, there
were no model-based assumptions in those values).
Common Experiments in AP Physics 1 (most you did, some were demonstrated)
See if you can use steps a)-e) above to design experiments for each of the labs below. On the following pages,
a design is outlined for each lab that follows steps a)-e). At the end is a question with an AP-style experiment
for you to design.
1. Determine the velocity and acceleration of a uniformly accelerating object
2. Determine the launch speed of a projectile shot from a launcher
3. Experimentally determine acceleration due to gravity, g
a) Free fall expt
b) Using an Atwood machine
c) using a Car on incline
d) Using Simple harmonic motion of a pendulum
4. Verify Newton’s 2nd Law using a modified Atwood setup
5. Experimentally determine coefficients of friction
6. Experimentally determine the speed of an object moving in UCM
7. Determine the acceleration of an elevator
8. Conservation of Momentum – NEED TO PUT THIS IN (also what happens to E)
9. Experimentally determine a spring constant (and/or determine whether a rubber band behaves like a
Hookean spring)
10. Experimentally determine the speed of a bullet using a ballistic pendulum
11. Determine the mass of a meterstick (or an unknown mass) using static equilibrium (given a known mass and
a ruler)
12. Experimentally determine the moment of inertia of an object
13. Determine the mass of a penny using the Atwood machine
14. Determine the speed of sound using a column (O/C or O/O), a tuning fork and a meterstick.
15. Determine the resistance of a circuit element (and show whether it is ohmic or nonohmic)
Experiment 1: Determine the velocity and acceleration of a uniformly accelerating object
Determining Velocity
a) MODEL: we expect vav=x/t
b) Select variables: Guided by the model, vav could be determined by measuring x (position) and t (time)
from a starting position and time.
c) Experimental Design: Collect at least 6 x-t points by
measuring position with a meterstick and time with a stopwatch
OR by measuring x and t with a motion sensor
OR by measuring v directly with a photogate
OR by measuring x and t with a video that can be analyzed frame by frame
d) Analysis: Plot x vs t (t is always on horizontal axis whether it is the independent variable or not)
Compare experimental graph to expected model: slope between two points, x/t is vav between those
points.
If v is constant, then the data is fit to a line and the bestfit slope is the experimentally determined
velocity.
If v is not constant, the x-t graph will not have a constant slope, will be curved. vav can then be
determined between each pair of successive points to plot a v-t curve.
e) Error Analysis: Compare to a known speed to find %error.
Determining acceleration of a uniformly accelerating object
There are a couple common ways to determine acceleration
Measure velocity
a) MODEL: we expect that a=aav=v/t
(or v=v0+at)
b) Select Variables: Guided by the model,
a could be determined by measuring v
and t.
c) Experimental Design: Collect at least 6
v-t points: Measure v and t with a
motion sensor
OR measure v and t by recording a
video that can be analyzed frame by
frame
d) Analysis: Plot v vs t (t is always on
horizontal axis whether it is the
independent variable or not).
Fit to a line (from model expect a linear
relation betw v and t).
Compare model and bestfit:
Slope of the line, v/t represents
experimentally determined a.
y-intercept represents initial velocity, v0.
Measure position
a) Model: For uniform acceleration, we expect that
x=x0+v0t+½at2.
b) Select variables: Guided by model, a could be determined
by measuring x and t.
c) Experimental Design: Collect at least 6 x-t points:
Measure position with a meterstick and time with a
stopwatch
OR by measure x and t with a motion sensor
OR by measure x and t with a video that can be analyzed
frame by frame
d) Analysis: Plot x vs t (t is always on horizontal axis
whether it is the independent variable or not).
Fit plot to a quadratic function (because expected x-t is
parabolic).
Compare the experimentally determined best fit equation,
x=A+Bt+Ct2, to the expected model (x=x0+v0t+½at2) to
determine what the bestfit coefficients, A, B, and C,
represent. From the comparison, x0, v0 and a can be
determined.
(Alternatively, if x0 and v0 are 0, could plot a linearized
graph of x vs t2. Slope of the x-t2 graph can be used to
determine a (slope of x-t2 graph is ½a))
e) Error Analysis: Compare experimentally determined a to a known acceleration to find %error. Systematic
errors that cause %error in come from assumptions made and from systematic experimental errors:
- One notable assumption made in the model-guided analysis is that acceleration is uniform.
Experiment 2: Determine the launch speed of a projectile shot from a launcher (using just a meterstick)
a) Model: Easiest to measure launch speed from a horizontally launched projectile because the launch
velocity only has one component (vx). Expected x=vxt so could measure x, (the range) and t (flight
time). However, if t is small, it’s difficult to measure t without lots of uncertainty. Instead could measure
y because it is also related to t: y= ½gt2. Combining the two, x=vx(√(2y/g))
Assumptions in the model: used kinematics assuming that the projectile is only under influence of gravity
(air resistance and was neglected)
b) Select Variables: Guided by model, launch speed, vx, could be determined from the horizontally launched
projectile by measuring x (range) and y (height).
c) Experimental design: Horizontally launch projectile and measure x (range) and y (height) with a
meterstick. Repeat a few times taking the average x (because it would have significant variability or
uncertainty).
d) Analysis: Use the model to guide the analysis – in other words, plug the data (xav and y) into expected
equation to determine an average launch speed, vx.
e) Error Analysis: Compare to an accepted speed (found by directly measuring the speed with a photogate
attached to the launcher) to find %error. Systematic errors that cause %error in launch speed come from
assumptions made in the model and from systematic experimental errors:
- One notable assumption made in the analysis is that the projectile is only under the influence of gravity
ay = g and ax = 0). In reality, there is air resistance (ax<0 and ay<g) and not accounting for it causes the
determined launch speed to be systematically overestimated (larger than accepted).
Experiment 3: Experimentally determine acceleration due to gravity, g
There are lots of ways to measure g
1. Determine g in a free fall experiment (see Experiment 1: determining acceleration of uniformly
accelerating object)
2. Determine g using an Atwood machine
a) Model of the Atwood Machine: Apply Newtons laws to the Atwood machine to find expected Atwood
acceleration Fnet syst/Msys = aCM =(m2-m1)g/(m1+m2). Assumptions in the model – the pulley is
massless (no rotational inertia)
b) Select variables: Guided by the model, g could be determined by measuring a and the masses, m1 and m2.
c) Experimental design:
m1 and m2 : measure with a balance
a: cant measure directly but because the forces and acceleration are constant, can measure distance a
block falls, y, (starting from rest) and the time it takes, t (with a stopwatch), and use those to determine
a (y= ½at2 assuming no air resistance). (See measuring acceleration in Experiment 1 for other ways
to measure a)
Repeat several times to get aav (or measure a for at least 6 different m1,m2 sets).
d) Analysis: Use the model to guide the analysis – in other words, plug the data (aav, m1, m2) into the
expected equation to determine an average value of g. (or if a was measured for several m1,m2 values,
plot a vs (m2-m1)/(m1+m2) and fit to a line. Slope of line is g)
e) Error Analysis: Compare experimentally determine g to 9.81m/s2 to find %error. Systematic errors that
cause %error in g come from assumptions made and from systematic experimental errors
- A notable assumption made in the model is that the pulley is massless (no rotational inertia). In
reality, it has mass and rotational inertia and not accounting for it causes the system mass to be
systematically too small. Using the model, g was found as the slope of the graph:
g=a(m1+m2)/(m2-m1). a and m’s were measured not calculated with the model so the model-
based assumption does not affect those values. The system mass is really bigger than m1+m2 and
so the assumption of a massless pulley leads to a slope that is an underestimate of g (smaller than
accepted value).
- Minor error is in the assumption of no air resistance in calculating a. That would lead to a small,
minor overestimate in a and g because the air resistance was minimal.
3. Determine g using a car accelerating down an incline (See Example 2 in the first section)
4. Determine g using simple harmonic motion of a pendulum
a) Model of the pendulum swinging back and forth: Assuming the pendulum acts like a simple harmonic
oscillator, expected period: T = 2√(L/g). Assumptions in the model – no energy is lost as the
pendulum swings (ideal massless string, no air resistance) and the angle of swing is small.
b) Select variables: Guided by the model, g could be determined by measuring T, period, and L, length of
pendulum.
c) Experimental design:
L (Indep.Var): measure with a meterstick
T (Dep.Var): measure with a stopwatch (because T is short and subject to lots of uncertainty, minimize
error by measuring 5T to determine T)
Collect at least 6 T-L pairs of data. Measure T for at least 6 different L
Controls and constants: mass of pendulum bob, starting amp kept constant.
d) Analysis: Plot T vs L (y vs x) and fit to a power function (because it is expected that T depends on √L).
Compare the experimentally determined best fit equation, T=AL0.5, to the model (T = 2√(L/g)) to
determine g.
Alternatively, could plot a linearized graph of T vs √L or T2 vs L. Slope of the linearized graph can be
used to determine g.
e) Error analysis: Compare experimentally determined g to 9.81m/s2 to find %error. Systematic errors that
cause %error in g come from assumptions made and from systematic experimental errors:
- One notable assumption made in the analysis is that the pendulum exhibits SHM.
Experiment 4: Verify Newton’s 2nd Law (a ~ Fnet and a ~ 1/m)
a) Model: Newtons 2nd Law: a = Fnet/M
Experimental setup: modified Atwood machine - system of car (mcar)+onboard mass (m) on smooth track
attached to a hanging mass, mh; total mass of system = M. Applying Newtons 2nd Law to the experimental
situation: Fnet = mhg (if friction between car and track is neglected)
b) Select variables: It is expected that a is directly proportional to Fnet (when M is constant) and that a is
inversely proportional to mass (when Fnet is constant). Measure acceleration of an object or system in
response to Fnet and mass.
c) Experimental design:
Part 1: Measure a vs Fnet (keeping M constant)
Fnet (Indep.Var): Fnet is provided by weight of
hanging mass (=mhg). Measure mh with balance.
a (Dep,Var): measure v-t with motion sensor; slope of
v-t is a (also see Experiment 1 for other ways to
determine a)
For system of car+onboard mass on smooth track
attached to a hanging mass, mh, measure mh, mcar
and m (sum is M). Total M must remain constant
Collect at least 6 a-Fnet pairs of data by measuring a
for at least 6 different values of Fnet (different
amounts of hanging mass). In all 6 measurements,
keep total system mass, M, constant by moving
mass from car to hanging mass so that mass of
accelerating system is constant.
Part2: Measure a vs M (keeping Fnet constant)
M (Indep.Var): Measure total accelerating mass
(mcar+m+mh) with balance.
a (Dep,Var): measure v-t with motion sensor;
slope of v-t is a (also see Experiment 1 for
other ways to determine a)
For system of car with onboard mass on smooth
track attached to a hanging mass, mh, measure
mh, mcar and m (sum is M). mh (mhg=Fnet) must
remain constant
Collect at least 6 a-M pairs of data by measuring
a for at least 6 different values of M (different
amounts of onboard mass). In all 6
measurements, keep Fnet constant by keeping the
hanging mass constant.
d) Analysis:
Part 1: Measure a vs Fnet (keeping M constant)
Plot a vs Fnet (y vs x)
Fit to a line (because it is expected that a is directly
proportional to Fnet).
Compare the experimental bestfit to expected model (a =
Fnet/M):
Slope of bestfit (a/Fnet ) represents 1/M, the inverse of
the system mass
y-intercept of bestfit represents error. Expect that a=0 at
Fnet=0. But that is assuming that the hanging mass
were indeed equal to Fnet (we did not make any
assumptions about a, that was measured). However
because the friction between the car and track was
neglected, the graph does not show a 0 y-intercept (see
graph below). Rather a = 0 when there is a small +Fnet
which shows that the hanging mass is balanced by
another neglected force – friction between car and track.
Part2: Measure a vs M (keeping Fnet
constant)
Plot a vs M
Fit to power function (because it is expected
that a is inversely proportional to M when
Fnet is constant).
Compare the experimentally determined
best fit equation (a=kM-1), to the model (a =
Fnet/M), Newtons 2nd. Comparison shows
that the bestfit coefficient represents the
experimentally determined Fnet exerted by
the hanging mass. Alternatively, could plot a
linearized graph of a vs 1/M; slope of the
linearized graph can be used to determine the
constant Fnet.
See graphs below
e) Error analysis: Compare the experimentally determined system mass to the accepted and directly
measured mass to determine %error. Systematic errors that cause %error in mass come from
assumptions made and from experimental systematic errors:
- Notable assumption made in this experiment was that there is no friction between car and track and
that the hanging mass was the net force. In reality, there is friction and not accounting for it causes
Fnet to be systematically too large which causes the experimentally determined total mass, M to be
overestimated (acceleration was directly measured, no assumptions made)
- Possible experimental systematic errors – track was not horizontal which would lead to a
measurement of a that was systematically too large or too small.
Experiment 5: Experimentally determine coefficients of friction
1. Coefficient of kinetic friction (See Example 1 in the first section) (there are many ways to determine k
based on applying models to various experimental situations. For example, see if you could design an expt
to determine the coefficient of kinetic friction between a block and an incline when block is sliding down
using either Newtons laws or energy to model the situation)
a
FnetNeglected friction
a
Msys
a
1/Msys
OR
2. Coefficient of static friction: Below are a couple common ways to s
Pulling Method
a) Model: It is expected that fs ≤ SFN
b) Select variables: Guided by the model,
could determine S, by measuring max fS
and FN.
c) Experimental Design
FN (Indep var): FN = mg if pull block along
horizontal surface
max fs (Dep var): cant measure directly, but
if pull block with spring scale while it
remains at rest, then Fpull = fk
Measure weight of block (=FN). Using a
spring scale or force meter, Apply a
gradually increasing horizontal force to an
object AT REST until it just starts to move.
Measure the max Fpull (= fS) to remain at
rest.
Measure fS max for same object with at least 6
different added masses (6 different FN).
d) Analysis: same as Example1 in 1st
section.
Angle of Repose Method – determine s (max value) of
block at rest on an incline (using just a meterstick)
a) Model: By applying Newtons Law to the experimental
situation (all forces balanced, a=0), one gets the model
s = tan ( is incline angle). (The angle of repose, ,
is defined as the incline angle at which an object just
starts to slide down an inclined plane; in other words, it
is the incline angle where Fgx = fs max.)
b) Select variables: Guided by the model, s could be
determined by measuring , the angle at which the block
just starts to slide (angle of repose).
c) Experimental design: Place block at rest on an incline
and slowly increase the incline angle until the object just
begins to slide. Measure that angle of repose, , with a
protractor OR determine by measuring the height of
the block and its distance along the incline to the ground
(sin=h/d). Repeat several times and average
d) Analysis: Use the model to guide the analysis – in other
words, plug the data (av) into expected equation to
experimentally determine s.
e) Error analysis: Compare the experimentally determined coefficient to an accepted value if available to
determine %error
Experiment 6: Experimentally determine the speed of an object moving in UCM
For a conical pendulum like the flying pigs, there are two ways to determine the speed of the pig:
Method 1
a) Model: By applying Newtons 2nd Law to the conical
pendulum, one can get an expression for the speed:
v = √(grtan) where is the angle the string makes with the
vertical. Assumptions made in the model: pig flies in UCM
which is a pretty good approximation
b) Select variables: Guided by the model, to determine v, we
can measure the r and of the flying pig.
c) Experimental design:
r: it’s very difficult to accurately measure r because the middle
of the circle needed to measure r is not visible or tangible.
Instead, measure d, the diameter of the circle (r=d/2).
: it’s very difficult to directly measure because the vertical
line needed to measure is not tangible. Instead, measure L,
length of pendulum and d, the diameter of the circular flight
(sin=r/L).
Method 2
a) Model: For UCM, sav=v =d/t = (2r)/T
b) Select variables: Guided by the model, v
could be determined by measuring r and
T of the flying pig.
c) Experimental design:
r: it’s very difficult to accurately
measure r because the middle of the
circle needed to measure r is not visible
or tangible. Instead, measure d, the
diameter of the circle (r=d/2).
T: measure T with a stopwatch.
Repeat measurements with high
variability several times and take
averages
d) Analysis: Use the model to guide
Repeat measurements with high variability several times and
take averages.
d) Analysis: Use the model to guide the analysis –plug the data
(rav and av) into expected equation to experimentally
determine v.
analysis – plug the data (rav and av) into
expected equation to experimentally
determine v.
e) Error analysis: Compare the two experimentally determined v’s to determine %difference Systematic
errors that cause %difference come from assumptions made and from experimental systematic errors
Experiment 7: Determine the max acceleration of an elevator
a) Model: Apply Newtons 2nd Law to an object of mass m in an elevator; it is expected that
FN = mg ± ma where a is the acceleration of the elevator (it is + if a is positive, - if a is negative).
b) Select Variables: Guided by the model, to determine a, can measure FN and m.
c) Experimental design:
m: measure using a balance
FN: Measure using a bathroom scale, force plate, or measure m with a balance (FN=mg)
Measure m of an object while at rest. Take same object and measure max/min FN as elevator accelerates.
d) Analysis: Use the model to guide analysis – plug the data (m and FN) into expected equation to
experimentally determine a.
e) Errors Analysis: Compare experimentally determined elevator acceleration to an accepted value to find
%error.
Experiment 8: Experimentally determine the spring constant of a spring (or a rubber band).
There are two ways to determine the spring constant experimentally
Static Method
a) Model: For an ideal (Hookean) spring, it is expected
that |FS| = k|x|.
b) Select variables: The spring constant, k, could be
determined by measuring FS and x.
c) Experimental design:
x (Indep.Var): stretch spring horizontally with a
spring scale or vertically with a hanging weight.
Measure x from equilibrium with a meterstick.
Fs (Dep.Var)
i. Horizontal spring – using a force meter or spring
scale, apply a pulling force to stretch the spring to
x and hold at rest so that FS = Fpull.
ii. Vertical spring – using hanging mass, apply vertical
force to stretch the spring x so that FS = mg
Collect at least 6 FS-x pairs of data
d) Analysis: Plot FS vs x and fit to a line (because it is
expected that FS is directly proportional to x).
Compare the experimental bestfit to model (|FS| =
k|x|)
Dynamic Method (mass oscillating on a spring)
a) Model: An object of mass m on an ideal spring
will oscillate in SHM when pulled from
equilibrium and released. It is expected that
the period of oscillation is TS = 2√(m/k).
Assumptions made with this model – there is
no energy lost as the mass oscillates back and
forth and the spring obeys Hooke’s Law
b) Select variables: Guided by the model, k
could be determined by measuring TS and m.
c) Experimental design: For at least six different
masses m (indep variable, measured with a
balance) measure S (dependent variable,
measured with a stopwatch).
d) Analysis: Plot T vs m and fit to a power
function (because it is expected that T depends
on √m). Compare the experimentally
determined best fit equation, T=Am0.5, to the
expected equation (TS = 2√(m/k)), one can
determine what the coefficient of the best fit
Slope of the line, FS/x represents k, the spring
constant.
NOTE: the FS-x plot is NOT linear for elastic
materials like rubber bands. A rubber band is not
“Hookean”; therefore, the elastic force it exerts is not
a restoring force.
equation represents and use that to determine k.
Alternatively, could plot a linearized graph of
T vs √m or T2 vs m. Slope of the linearized
graph can be used to determine k.
e) Error analysis: Compare experimentally determined k to accepted value if available to find %error.
Systematic errors that cause %error come from assumptions made and from experimental systematic errors
Experiment 9: Experimentally determine the speed of a bullet/ball using a ballistic pendulum
a) Model: In order to derive a model or expected equation for the speed of a launched ball, v0, the appropriate
physics needs to be applied to the experimental situation. Conservation of momentum (of the
ball/pendulum system) can be applied during the collision and conservation of energy (of the
ball/pendulum/Earth system) can be applied during the swing of the pendulum. Energy (of b/p/E)is NOT
conserved during the collision (it’s perfectly inelastic) and momentum (of b/p) is NOT conserved during
the swing (as the bullet/pendulum swing up, they are losing momentum to the earth because the pendulum
is fixed in place). By separating the process into parts (collision and swing) and using Cop and CoE
appropriately, you can derive an expected equation for the speed of the ball before collision in terms of the
max height the pendulum rises to: v0 = ((mball + Mpend)/mball)(2gh) where h is the max height pendulum
swings to after collision (you should be able to derive this).
b) Select Variables: Guided by the model, v0 could be determined by measuring the masses of the ball and
pendulum and the max vertical height of the pendulum after collision h.
c) Experimental design: .
mball and mpend : measure with a balance.
h: It’s very difficult to accurately measure h directly because it is not a tangible height and you would have
to do a very crude estimation. Instead, measure L, length of pendulum with a meterstick (Note: for the
ballistic pendulum with significant mass in the shaft, L is the distance to the CM and not to the bob) and
measure max, the max angle at max height (with an attached rotary sensor). From L and max, h can be
determined with trig (h=L-Lcosmax). Repeat max measurement several times to determine hav.
One way to measure h directly - take a video and analyze frame by frame. Repeat h measurement
d) Analysis: Use the model to guide analysis – plug the data (mball, mpend and hav) into expected equation to
experimentally determine v0 an average bullet/ball speed.
e) Errors Analysis: Compare to an accepted speed (found by directly measuring the speed with a photogate
attached to the launcher) to find %error.
Experiment 10: Determine the mass of a meterstick (given a known mass and a meterstick for
measurement)
a) Model: One can balance the meterstick (mass Mms) and a known mass, m, on a pivot (edge of table) so that
it is in static equilibrium. Applying Newtons 2nd Law to the rotational equilibrium, it is expected that the
sum of the torques must be zero: (Mmsg)rms+(mg)r = 0 where r is measured from pivot (r=0) and could be
positive or negative depending on the direction of the torque.
b) Select Variables: Using the model and given m, Mms could be determined by measuring the distances rms
(distance of meterstick CM to the pivot) and r (distance of m to pivot),
c) Experimental design: A meterstick and a known mass m placed on it at some position r, are placed on a
pivot (or the edge of a table) so that the system of meterstick and known mass is in static equilibrium.
Measure r values with a meterstick. Be careful to include whether r is + or – depending on the direction of
torque around the pivot. Setup the meterstick and mass system in several different equilibrium positions and
measure the r values.
d) Analysis: Use the model to guide the analysis . Shown are 2 ways to analyze
1. Plug the data (m, r and rms) into expected equation to experimentally determine Mms for each set of data.
Use average value of Mms.
2. Model: mr =Mmsrms. Plot r vs rms and fit to a line (because it is expected from the model that these
variables are linearly related) Compare experimental bestfit to model (r = (Mms/m)rms) in order to
determine what the slope and intercept of bestfit represent and to determine Mms. Comparison shows
that slope represents Mms/m. Intercept should be 0 (if it is very close to 0, it represents random
experimental error; if it is a significant value, it represents some systematic error)
e) Errors Analysis: Compare experimentally determined mass to the known value to find %error.
Extension experiment: Given a meterstick, a known mass and an UNKNOWN mass, determine the mass of
the meterstick (as above) and the UNKNOWN mass. See if you can design the experiment using steps a)-e) and
clearly writeup your experimental design.
Experiment 11: Experimentally determine the moment of inertia of a rotating object or system of
rotating objects.
There are two ways to determine the rotational inertia of an object
Experimental, Dynamic Method
a) Model: We can apply Newtons 2nd to a rotating system of objects:
net = Isys
b) Select variables: Guided by the model, the rotational inertia, Isys,
could be determined by measuring net and of a rotating object or
system of objects.
c) Experimental design: Apply various net (indep var) to an object
free to rotate by hanging various masses, m (measured with
balance), on a string wrapped around the axle at a distance rp from
rotation axis. Determine the resulting of the object (dependent
variable). (draw a diagram of the setup with all variables labelled).
Collect at least 6 pairs of net - data points.
(Dep.Var): cant measure directly. Instead, use an attached rotary
sensor to measure -t as object rotates (the slope of -t graph is
)
net (Indep.Var): difficult to measure directly because it consists of
the torques produced by the string tension and the torques due to
the friction of the rotary axle, pulley and other parts of the setup.
If friction of the axle and various pulleys that bend the tension
are neglected (an assumption that introduces error), then
net = T = rpT and this can be determined by measuring rp (radius
of axle string is wrapped around) and from the acceleration of
the falling mass: T = mg-ma = mg-m(rp). Then net can be
determined by measuring m, the falling mass (with a balance), rp
Theoretical Method
a) Model: The rotational inertia of a
system of objects rotating around
an axis can be calculated if the
rotational inertias of the
components of the system are
known theoretically. For example,
the rotational inertia of a system
containing a disk (ID= ½MDRD2)
and a ring (IR = ½MR(Ri2+ Ro
2) is
the sum of the individual I’s:
Isys = ID + IR
b) Select variables: Guided by the
model, Isys could be determined by
measuring the masses of each
object in the system (MD and MR)
and the relevant distances from the
rotation axis (R values)
c) Experimental design: measure the
masses with a balance and the R’s
with a meterstick.
d) Use the model to guide analysis –
plug the data (Ms and Rs) into
expected equation to determine the
net
Neglected torque due to friction
(radius of pulley string is wrapped around and (as described
above)
d) Analysis: Plot net vs (y vs x) and fit to a line (because it is
expected that net is directly proportional to ).
Compare the bestfit equation to the model (net = Isys).
Slope of the line, net/a is Isys, the rotational inertia of the rotating
system.
Y-intercept: The graph should show a direct relationship. A
significant non-zero intercept is caused by a systematic error. The
value of net was based on an assumption that net=T. However
because the axle and pulley frictions were neglected, the graph
does not show a 0 y-intercept (see graph below). Rather = 0
when there is a small +net which shows that the torque produced
by the tension is balanced by another neglected torque– friction of
the axle- that’s what the y-intercept represents.
theoretical Isys.
e) Error Analysis: There are not
many errors in this method, just
small random measurement errors.
Systematic errors would also be
very small if the disc and ring had
small scratches and nicks that
changed their true I a bit.
e) Error analysis: Compare experimentally determined I to the theoretically determined I (close to accepted)
to find %error. Systematic errors that cause %error come from assumptions made and from experimental
systematic errors
- Notable assumption made in this experiment was that there is no axel
friction torque In reality, there is friction and not accounting for it
causes net to be systematically too large which causes I to be larger
than the accepted value ( was directly measured, no assumptions
made)
- Another assumption was that the experimentally determined I was just
the I of the system of objects; it was assumed that other rotating parts
in the setup like the pulleys, added no I to the rotating system.
Therefore the experimentally determined I includes those unaccounted for I’s and is an overestimate.
Experiment 13: Determine the speed of sound using a tuning fork, a meterstick and a variable length
Closed-Open tube
a) Model: The experimental setup (shown at right) typically uses a tuning
fork that produces sound at one given frequency (instead of tuning
fork, can use a tone generator of constant frequency). A tube is closed
at one end by placing it in water; the length of the tube can be changed
by putting more/less of the tube in the water. When the tuning fork is
placed over the tube closed at one end, the sound will get loudest at the
length that produces resonance or a standing wave. Apply physics to
the experimental situation to make a model: The shortest tube length
that produces resonance is the fundamental and its length is L = ¼1.
The expected speed of sound in the tube is v = f = 4Lf = v.
b) Select Variables: Guided by the model, knowing the frequency of the tuning fork, v could be determined by
measuring L when there is resonance at the fundamental.
c) Experimental design: Use a tuning fork or tone generator of known frequency and increase length of tube
from 0 by raising it out of water until you hear resonance at 1st harmonic. Measure resonant length with a
V
I
nonohmic
ohmic
meterstick. Repeat several times and take average L.
More accurate way to do this experiment is to measure the lengths at 2 successive harmonics:
expected L = /2 = v/2f
d) Analysis: Use the model to guide analysis – plug the data (f and Lavr) into expected equation to determine
an average value of v, the speed of sound in air.
e) Error Analysis: Compare experimentally determined speed of sound to the known value (343m/s) to find
%error. One of the systematic errors in this experiment that was not accounted for in the model is that the
resonant wave extends out of the tube a bit and the amount it extends over depends on the diameter of the
tube. Neglecting this effect results in an underestimate of L and therefore a v that is less than the accepted
value. (by finding the resonance at 2 successive harmonics and L as described above, this error is
subtracted out)
Experiment 14: Determine the resistance of a circuit element (and show whether it is ohmic or
nonohmic)
a) Model: Each circuit element obeys Ohms Law V = IR
b) Select variables: Guided by the model, R could be determined by measuring V across and I through a
circuit element,
c) Experimental design: Connect a battery or voltage source across a circuit element (resistor or light bulb).
V: Measure, the voltage across the bulb or resistor with a voltmeter connected in parallel to the element.
I: Measure, the current through the bulb or resistor with an ammeter connected in series to the element.
Change the voltage across the circuit (add batteries or change voltage) to collect at least 6 V-I pairs of data
d) Analysis: Plot V vs I.
Compare the experimental graph to the model (V = IR)
Slope of the curve represents R, the resistance of the circuit element
If the V-I plot is linear and the slope is constant, then the circuit element
is “ohmic”; in other words, the resistance of the device is constant for a
range of currents and voltages (Resistors are ohmic for a range of
currents and voltages).
If the V-I graph is not linear, then the slope (R) is not constant and the
circuit element is “nonohmic”; in nonohmic devices such as lightbulbs,
the resistance usually increases with current through the device.
e) Error analysis: Compare experimentally determined R to an accepted value to find %error.
EXPERIMENT FOR YOU TO DESIGN: (the AP test has one experimental design free response question)
A student makes the hypothesis that the %energy lost when a ball bounces off the ground increases with the
height it is dropped from.
Using steps a)-e) design an experiment to test the student’s hypothesis.
1. Make a model of the experiment with a clearly labeled diagram
2. Describe the procedure
- what variables will be measured and how will each be measured
- which variable is indep, which is dep
- Give a clear and concise step by step procedure.
3. Explain how the data will be analyzed:
- What will be plotted?
- How will the graph be analyzed to address the hypothesis?
- What will the graph look like if the data supports the students hypothesis?
4. What are the main sources of systematic error in your experiment that would cause %error?
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