Page 1 of 21
Physics A.P. C Lab Manuel 2011 (Mechanics)
Instructor Mr. Farrell
Page 2 of 21
Table of Context
Lab
Unit
Page
1.Great America Lab
Review Mechanics
2.
2.Finding limits
Ch 1 (Introductory Calculus
4.
3.Hitting a moving Target
Ch3.Projectile Motion
(Kinematics Equations)
6.
4.Coffee Filter Lab
Ch 4.Drag (Air resistance)
7.
5. Work Energy Theorem
Ch 5“Roller Coaster”
(Energy diagrams)
9.
.
6. Air Tracks (Elastic vs Inelastic) Ch 6. Momentum and Impulse 11.
7. Rocket Science
Ch. 7 Changing mass,
12
(conservation of momentum)
8. Toilet paper race
Ch 9.Rotational motion
17.
10. Finding the moment of Inertia Ch 11. Torque/Angular momentum 18
of a rotating wheel
11.Calculating the spring constant Ch 12. Simple Harmonic Motion 20
of a spring
Page 3 of 21
Great America Field Trip
Purpose :
#1To Review the principle of mechanics learned in first year physics
#2. Apply physics concepts to real world experiences
#3. Become familiar with Advanced Placement Standards.
A.P. Standards (1st Semester)
I.
Newtonian Mechanics
A. Kinematics
2. Motion in two dimensions
1. Displacement
2. Velocity
3. Acceleration
B. Newton’s law of motion
1. Static Equilibrium (First Law)
2. Dynamics of a single particle (Second Law)
3. Systems of two or more particles (Third Law)
C. Work, Energy, Power
1. Work and the Work Energy Theorem
2. Forces and Potential Energy
3. Conservation of Energy
4. Power
D. Systems of particles, Linear momentum
1. *Center of Mass
2. Impulse and Momentum
3. Conservation of Linear momentum, collisions
E. Circular Motion and Rotation
1. Uniform Circular Motion
2. *Torque and Rotational Statics
3. *Rotational Kinematics and dynamics
4. *Angular Momentum and its conservation
F. *Harmonic Motion
*(Material not included in lab)
Page 4 of 21
Procedure :
You will need to carry out at least three experiments at the park to investigate the physics
principles listed in the A.P. standards. These standards you should be familiar with from
enriched physics. The rides you decide to perform the experiments on are up to you, however
the experiments must investigate at least three physics standards stated above.
These five steps must be completed before the day of the lab.
#1. Choose the concepts you want to study. (At least Three)
#2. Pick the ride that you would like to do the experiment on. Make sure that it is a ride you or
your partner are willing to go on.
#3. List the equipment you will need
#4. Write up a procedure describing how you will be carrying out your experiment. Be as
detailed as you can, listing each step in the process.
Be sure you list the equipment you will be using and explain how it will be used.
#5. Develop a data table stating the control, independent and dependent variables.
Materials that will be available to you
Stop Watch
Tape Measure
Accelerometer (Pasco Equipment)
Altimeter
Results/ Graphs and Analyses
Generate appropriate graphs showing the relationships between dependent and independent
variables.
Examples: displacement – time, velocity – time, Force- time
Explain what the graph tells us about the motion of ride over time.
Below is a list of WEB sources that will help you in your preparation
http://Homepage.mac.com/cbakken/cga/curriculum/
http://www.marshall.k12.wi.us/mhs/teachers/mirus/20002001/GreatAmerica/GiantDrop.html
http://www.marshall.k12.wi.us/mhs/teachers/mirus/20002001/GreatAmerica/GAMindBoggle.ht
ml
Lab 2 Instantaneous Velocity Lab
Page 5 of 21
Objective – To understand how finding the limit of a function as time approaches zero results in
finding the instantaneous solution for that function
Purpose: To determine the instantaneous velocity of a cart that starts from rest and accelerates
down an air track at some chosen distance and angle.
Key Terms: Position, displacement, Speed, Average Velocity, Instantaneous velocity, Limit,
derivative.
Essential questions: How are velocities measured experimentally? (Ex. car, baseball, and
sprinter)
What challenges are present in determining an objects instantaneous velocity through
experimentation?
Theory: The average velocity of an object is defined as the change in an objects position over
time. If the starting position is X0 and starting time is 0s we a have:
The instantaneous velocity v(t) is defined as
Where x(t) is function that represents the position of an object at time t and X(t+dt) represents
the position of that object a short time later. The change in position over the change in time
interval is the average velocity for that time interval. As the time interval approaches zero the
average velocity approaches the instantaneous velocity.
V(t) = V(ave)
X
30
Page 6 of 21
Materials:
Photogates
Cart
Track
Motion Timer
Procedure: Write a procedure that allows you to find the relationship between the average
velocity as a function of decreasing time interval. Describe how your experiment minimizes
error and describe any assumptions that you make, such as the acceleration is g sine A.
Results/Graph Analyses
1. Calculate the average velocity (V) for all trials and record.
2. On Regular graph paper or graphing program graph the average velocity as a function
of time.
3. Extrapolate best fit curve to the y axis to find the instantaneous velocity. The velocity
when time between intervals equals 0.
4. Calculate the theoretical value for instantaneous velocity using the equation
5. Perform an error analyses
Determine percent error, standard deviation, and describe in detail the experimental
errors and how you tried to alleviate them.
Page 7 of 21
Projectile motion lab
Purpose:
a. To graph the motion of a projectile in terms of its range and launch angle.
b. Develop and apply the kinematic equations for two dimensional projectile motion.
Theory: A projectile is an object in two dimensional motion under the influence of gravity. The
objects trajectory is the path in which the object takes. The range is the horizontal distance the
object travels. The motion of a projectile can be separated into two problems a) Constant motion
in the x direction. b) Free fall in the y direction. The projectile’s velocity first must be separated
into its components horizontal Vx = Vcos A and vertical Vy = Vsin A before further analyses
can be performed.
Directions:
You are required to set up and perform an experiment to map out the trajectory of a Pasco
Projectile Launcher. The initial velocity will be constant per setting so by varying the angle of
trajectory and measuring the range (horizontal distance) you can calculate the initial velocity of
the ball. The experiment needs to be able to collect precise data which will be used to make
predictions with in an uncertainty of + or – 1cm. You will have one day to perform the
experiment, collect data, and analyze it. From the results of your experiment you will be asked
to hit three unknown positions.
#1. Horizontal displacement 150 cm and a vertical displacement 0.
#2. Horizontal displacement 150 cm and the vertical displacement 150 cm.
#3. Vertical displacement 0 and a horizontal displacement of (50cm + vt)
v - velocity of the moving target
t – time the object is in the air.
Hitting target one will give you a 80%
Hitting target two will give you a 90%
Hitting target three will give you a 100%
Page 8 of 21
Ch 5The Coffee Filter and Air Resistance
Introduction
In enriched physics you always made the assumption that the air resistance was negligible. However,
drag has an significant effect on most falling objects If an object of mass M falls under the influence of
gravity and a drag force Fdrag we may write Newton’s Second Law as:
Mg – Fdrag = Ma
(1)
Here a is the acceleration of the object, and Mg is the weight of the object. We may model the
drag force Fdrag as shown in Equation 2
Fdrag = bVn
(2)
Here the drag coefficient b is a constant that depends on the shape of the object, and V is the
velocity of the object. For objects moving at low velocity through air n is usually close to 2. By
substituting Equation 2 into Equation 1 we obtain:
Ma = Mg – bVn
(3)
As an object falls from rest the velocity increases until the drag force and the weight are equal in
magnitude. The acceleration is then zero, and we have
Mg = bVn
(4)
This may be rearranged to give the terminal velocity Vf:
Vf = (Mg/b)1/n
(5)
Now by taking the logarithm of both sides of Equation 5 we obtain
ln(Vf) = (1/n) [ln(M) + ln(g) – ln(b)]
(6)
This may be rewritten as
ln(Vf) = (1/n) ln(M) + C
(7)
where C is a constant equal to (l/n)[ln(g) — ln(b)]
We can now determine the coefficient n of the velocity V if we can determine the terminal velocities V~
for a variety of masses. A log-log plot of the terminal velocity versus the mass then yields a linear plot
with a slope of 1/n.
The problem with varying the masses is that the drag coefficient depends on the size and configuration
of the object. This is why we can’t do the lab by dropping a ping-pong ball, a golf ball and a bowling ball
from the top of the school building. This is where the coffee filters come in. By dropping stacks of coffee
filters consisting of one, two, or more filters, we may vary the mass of the falling object while keeping its
size and configuration approximately the same.
Page 9 of 21
Experiment
The coffee filters used in this lab are those that look like an inverted crown or a basket. The corrugations
of the filter make it easy to nestle them one inside the other. To carry out the lab, a large vertical drop
of a few meters must be used. The best results are acquired by using as large a vertical drop as feasible.
Procedure
One student drops the filters, while a second student times their fall. The average velocity V of the filters
is found by dividing H, the height of the drop, by T, the time of the fall from release to impact. Heights of
at least 3 meters be used for H. Several (5-10) trials should be used for each value of the mass, or
number of filters.
You may perform the experiment by dropping the coffee filters in multiple of 3 or 4 this will allow you to
see the impact that mass has quicker.
Graph the data on log-log paper, with velocity on the vertical axis and the number of filters on the
horizontal axis. What is the resulting slope of the straight line is equal to?. Now graph the velocity on
the vertical axis and the fall time on horizontal axis, What does the slope equal now?
Assumption: The average velocity given by H/T is equal to the terminal velocity.
Is this a good assumption? Why or Why not?
What effect does this approximation have on the answer for n?
Page 10 of 21
Ch 6 and7 (Work Energy theorem)
Theory: To reduce the complexity of the derivation, we will assume that the direction of the Net
Force is constant while the work is being done. The Work-Energy Theorem is still valid if the net
force changes direction as well as magnitude while the work is being done, provided the body is
rigid.
The key step is to convert the calculus definition for acceleration into an expression that is a
derivative of x.
Plug this and the Second Law into the definition for Work, and integrate.
Conservative Vs. Nonvonservative forces.
A force is conservative if the work it does moving an object between two points is the same for
any path taken between the two points.
The work done by a non-conservative force does depend on the path taken.
Purpose: To explore the principle of conservation of mechanical energy for a rolling object.
To start to understand the concept of moment of Inertia.
Pre lab Questions
Do you think a rolling object conserve Mechanical energy (Mgh1+ KE1 = Mgh2 + KE2)?
Is the work energy theorem valid for this situation?
What are the forces acting on the metal ball as it rolls down the coaster?
Are the forces conservative or non-conservative?
Page 11 of 21
What do you think the diameter of the rolling metal ball affect the apparent loss of energy?
D
A
E
C
B
Directions
a. Calculate the potential energy (mgh) for five different heights on the rollercoaster.
b. Release the ball from the hieghest point and measure the velocity at each of the five points
c. Calculate the kinetic energy (1/2 m v2) at each of the five points
d. Compare the change in kinetic energy to the change in potential energy between at least six
different locations.
Location
Potential Energy
(mgh)
Velocity
Kinetic Energy
1/2mv2
Total energy –
Mgh +
1/2mv^2
A
B
C
D
E
Discussion Questions
1. From your results compare the total kinetic energy to the change in potential energy. Is
Mechanical energy conserved? If not where did the energy go?
2. How would the results change if the track was frictionless? Would the ball roll?
3. Would happen to the velocity of the ball if the ball slid instead of rolled?
How would your results for the change in kinetic energy change?
Page 12 of 21
Elastic Vs Inelastic Collisions
Theory: Elastic collisions conserve energy and inesatic collisons do not, but both types of
collisions conserve momentum. Momemtum of an object is its mass times velocity. If there is
no outside force on a system of bodys during a collision the total momentum before that collision
will be equal to the total momemtum after the collision. The change of momentum is also called
impulse which is equal to the integral of force as a function of time.
Experiment
There are two different rubber ends to the wooden pendulum you will be using to make a
collision with the the cart. One end bounces off the cart with a minumum loss of energy and the
other end hits and moves with the cart inesasticly.
Purpose: Conduct an experiment measureing the impulse of both the inesatic and inesastic
collison. Is their a way to determine the Average applied force
Hypothesis: Predict a whether the elastic or inestastic collision will have the greater impulse or
whether impulses will be the same. Which one do you think will have a greater contact time?
Which will have the greater force applied?
Is there a way to determine the impulse from a Force Vs time graph?
Page 13 of 21
Data Colliection
Release
Height
Total
Energy
mgh
Elastic Collsion
Mass of
Car
Initial
Velocity
Final
Velocity
Average
Force
Applied
Time of
Impact
Final
Velocity
Average
Force
Applied
Time of
Impact
Inelatic Collsion
Release
Height
Total
Energy
mgh
Mass of
Car
Initial
Velocity
Page 14 of 21
Ch 8 Rocket Science
Safety Note: Keep away from the rocket engine
during ignition. Wear safety goggles to protect your
eyes. Ignite rocket from a distance, using the long
electrical wires provided. If you’re not sure how to
fire the rocket safely, ask the instructor for guidance.
Introduction:
The purpose of this lab is to test concepts of rocket flight with an Estes Alpha III
rocket powered by a B6-6 or C6-7 engine. The B6-6 exerts an average force of 6N for
0.83 s, for a total impulse = 5 Ns. The C6-7 exerts an average force of 5.9 N for 1.70 s,
for a total impulse of 10 Ns. During the thrust phase, the rocket accelerates. After the
thrust phase, the rocket continues to climb, reaches a maximum height, then starts to fall.
After a delay of several seconds, measured from the end of the thrust phase, a parachute
is deployed to prevent damage to the rocket or ground-based observers on impact. For the
B6-6 rocket, the delay is 4 s; for the C6-7, the delay is 7 s. A diagram of the rocket’s
height vs time thus looks like this:
Operationally, the lab consists of two parts:
1. Using a Pasco force sensor, measure the force-time curve of a B6 or C6 rocket engine.
2. Launch the rocket and
a) measure the maximum height reached by rocket when powered by the same
type of engine and compare to predictions to the real results.
b) from a video recording of the launch, measure the initial acceleration of the
rocket and compare the predicted and experimental results.
Questions you need to answer in your lab report, neglecting air
resistance and the change in mass of the rocket during launch:
1. Will your rocket get off the launching pad?
2. What is the velocity of the rocket at the end of the thrust phase?
3. What is the maximum height reached by the rocket?
4.. How long does it take to reach its maximum height?
5. Does the parachute deploy before or after the rocket reaches its maximum height?
6. How does air resistance affect the results in 2, 3, 4, and 5?
7. Compute the velocity of the gases leaving the engine.
What measurements do you need to make to answer these questions? ( Think!! Make a
list!!) You are encouraged to work out the equations for parts 1-4 for yourself before
doing experiments. Otherwise, you will probably get lousy data. If you can’t work out
the equations yourself, some guidelines are provided at the end of this write up.
Experiments:
Page 15 of 21
Part I. Measure the impulse I and thrust F of the rocket engine using the Pasco Force
Sensor. Do this twice and average the results..
A. Weigh the engine before ignition
B. Set up the Pasco Force sensor with rocket motor attachment, Pasco Interface,
and your own computer to measure F vs t. Ignition is done OUTSIDE THE
BUILDING, ON THE LOADING DOCK, to avoid pollution of the building;
C. Put on Safety goggles.
D. Insert engine in Force Sensor with exhaust pointed AWAY FROM the sensor
and away from the building.
If the engine does not fire, WAIT A FULL MINUTE before
approaching the rocket. Then insert a new detonator and try again.
E. Record F vs. t during rocket motor burn. Caution!! Rocket engine
spews out VERY hot gases!!! You should be at least 2
meters away from the engine during the burn. Compute the
impulse of the engine and the velocity of the escaping gas.
I=
∫
Fdt
G. Weigh the burned-out rocket motor
Part II. Rocket flight
A. Weigh the rocket and measure its diameter (for drag estimate)
B. Estimate:
1. the height at burnout and the maximum height, neglecting drag and mass
changes.
2. the time at which you expect the rocket to reach its maximum height,
neglecting drag.
3. decide on a method for measuring the maximum height. (Hint: using the
data from part one, first estimate the maximum height of the rocket.)
C. Align launch rod so rocket will go
straight up;
D. Set up a video camera to record the launch.
E. Put on goggles.
F. Fire rocket. Measure hmax and tmax.
G. If you recover the rocket, repeat the experiment.
Part III.
A. Answer the questions in the introduction.
B. From your data from the force meter:
Numerically calculate the height, velocity, and acceleration of the rocket as a
function of time. (Assume a drag coefficient of 0.5) Plot these results.
C. Compare these results with your experimental measurements.
1. Maximum height of the rocket.
2. Height vs. time acquired from the video of the launch.
Does a different drag coefficient improve the results?
Page 16 of 21
APPENDIX.
Analytical approach:
Assume a rocket of initial mass m0 is powered by an engine which is turned on at t=0.
The duration of the thrust is tthrust, giving an impulse
0
tf
I = ∫ Fdt . At the end of the “burn”
the rocket mass = mf. Assume the rocket goes straight up. What are the height and
velocity of the rocket as a function of t? What is the maximum height?
Simple approach
Assume the rocket mass is constant. This is roughly true for our toy rocket, but is
not okay for launching earth satellites, ICBMs, etc.
Neglect air resistance.
Divide the flight into a thrust stage and a coasting stage.
A. Thrust Stage (0 < t < tthrust) starts with v = 0 and ends with v=vthrust. Newton
II gives
/ thrust F − mg = mdv dt Eq. 1
so
( ) thrust F − mg dt = mdv Eq. 2
Integrating, one obtains v(t)
( ) thrust F − mg t = mv Eq. 3
Integrating again
1()2
2 thrust y F mgt
m
= − Eq. 4
At the end of the burn stage, t=tthrust, y=yf, v=vf.
B. Coasting Stage. Only gravity acts on the rocket, so
/ net F = −mg = mdv dt Eq. 5
Integrating, one obtains
( ) thrust f −g t − t = v − v Eq. 6
Integrating again, one obtains
1 ( )( )2 ( )
2 f f f f y = −g t − t + v t − t + y Eq. 7
One can solve for ymax using Eqs. 6 and 7. However, energy considerations give an
easier approach. At the end of the burn, the rocket has energy E = K + U,
where U=mgyf. At ymax, in the absence of drag, the energy is unchanged but
K=0, so one can solve for ymax.
5
Numerical Integration
A rigorous approach requires consideration of drag and changing mass. A simple way to
include these factors is to integrate the equations of motion numerically in Excel. The
idea is as follows:
Start at t=0, v=0, y=0, m = mass of rocket + fuel
Evaluate Fnet = -mg + Fthrust + Fdrag
Increment t by dt
Evaluate v using Newton’s 2nd Law
dv = Fnet dt / m
add dv to previous v:
Evaluate y
Page 17 of 21
dy = v dt
add dy to previous y.
Reevaluate Fnet ; it differs because of changes in m and v. (Assume that
the fuel is burned at a constant rate.)
Increment t by dt
Repeat . . .
Rocket Equations Quick Reference
Equations for finding your rocket's peak altitude and motor delay.
K – Drag constant
p = ½ air density = 1.22 kg/m3
Cd = Drag coefficient
A = surface are of rocket
T = Thrust of engine
m = mass of Rocket
t = motor burn time
v = burnout velocity in m/s
y1 = Altitude at burnout
yc = Coasting distance
ta = Coasting time
Page 18 of 21
Toilet Paper race
Purpose : To see the effect that rolling has on an objects motion. To
introduce the concepts of moment of inertia, torque, angular
acceleration, angular velocity, and the parallax theorem.
Hypotheses: Two rolls of toilet paper are dropped at the same distance
and time however one is dropped so that it does not unroll and the other
is dropped so that it rolls as it drops.
In Class Demo: Before the class performs the experiment, predict
which one will hit the ground first. Explanation:
After the class has performed the experiment in groups will try to come
up with a way to calculate the separation distance of the two rolls of
toilet paper when dropped so that they will hit the ground at the same
time.
Important points to consider
The drop time for the unrolled roll is tr = (2H/g)1/2
The drop time for the rolled roll is tur = (2h/a)1/2
The trick is to find the linear acceleration of the toilet paper.
a = ar
The torque t = I a
The moment of inertia of the toilet paper is I= 1/2M(R^2 +r^2)
inner radius r and the outer radius is R
According to the parallel axis theorem In = I +MR^2
Page 19 of 21
Ch 9 and 10 (rotational motion)
Theory: The study of the behavior (kinematics) and the causes (dynamics) of an object under
rotation can be understood using Newton’s laws of motion. The analyses of a rigid body will
include studying the relationship between a rotational object’s velocity, acceleration, energy,
momentum and net force. A major difference between translational and rotational motion is that
under rotation an objects center of mass is not enough to describe its motion. The mass along
with its position with respect to the center of mass needs to be known. This relationship is found
by summing the individual components of an object’s mass multiplied by its distance from the
center of mass squared.
In this lab you will determine the moment of inertia of a bicycle while investigating the
relationship between the wheels rotational velocity, acceleration momentum, energy, and torque.
Purpose: The purpose of this lab is to investigate the properties of rotation while
measuring the moment of inertia of a bicycle wheel.
Procedure: Write a clear set of steps for each procedure.
Procedure A.
Procedure E
Procedure B.
Procedure F.
Procedure C.
Procedure G.
Procedure D.
Procedure I.
Page 20 of 21
Data:
Starting
height
Initial
Potential
energy
(P.E)
Period of
revolution
(T)
Angular
velocity
(w)
Linear
Velocity
(v)
K.E. of
the
Wheel
Moment
of Inertia
(I)
Angular
momentum
(L)
Angular
Acc.
Torque
.25 m
.5 m
.75 m
1m
1.25 m
1.5 m
1.75 m
2.0m
Discussion Questions:
1. Did the calculations for the moment of inertia vary with drop height? If so explain the
variation.
2. Why could you assume that the translational and rotation energy of the wheel of was the
same? If the wheel was allowed to move along the floor as it rotated would the rotational and
translational energy remain the same? If not explain what happened to the energy lost?
3. Explain how the angular momentum is related to the objects rotational energy. Show this
relationship using an equation.
4. Explain how the torque applied to the wheel affects the objects angular velocity, acceleration,
momentum and energy.
Page 21 of 21
Name _____________
Date _____________
Purpose Statement:
a. Determine the constant k of their spring from investigating the relationship between mass
and the period.
b. Investigate the effects that adding springs in parallel and series has on the constant k.
Hypotheses:
What type of graph do you predict when plotting the period as a function of its mass.
Procedure:
a.
b.
c.
d.
e.
Weigh and record the mass beginning with the smallest mass.
Put the mass on the hanging spring and record the period of revolution and record.
Repeat experiment for several different masses each time recording its period.
Graph the period as a function of mass
Add a spring in parallel and Repeat experiment
Discussion Questions:
1. What type of graph did you get?
2. Use the following equations to derive a relationship between period and the spring
constant
3. Describe what happens to the spring constant when the springs are added in series and
parallel. How do you explain the changes to k?
Conclusion:
Reflection