Vector Addition of Forces

Purpose: To study vector addition by:
1) Graphical means and by
2) Using components.
A circular force table is used to check results.

Equipment: Circular force table, masses, massholders, string, protractor, four pulleys.

Procedure: 1. Your instructor will give each group three masses in grams (which will
represent the magnitude of three forces) and three angles. Choose a scale of
1 cm = 20 grams, make a vector diagram showing these forces, and
graphically find their resultant. Determine the magnitude (length) and
direction (angle) of the resultant force using a ruler and protractor.

mass 1. 100g 0 degree
mass 2. 100g 335 degree
mass 3. 100g 270 degree

2. Make a second vector diagram and show the same three forces again. Find
the resultant vector again, this time by components. Show the components of
each vector as well as the resultant vector on your diagram. Draw the force
(vector) you would need to exactly cancel out this resultant.

                                           scaled vectors and their vector addition steps 1 and2

                                             results from calculating the angle at 180 degrees
                                               opposite from resulting angle of first 30 vectors
                                                (step 3)

3. Mount three pulleys on the edge of your force table at the angles given
above. Attach strings to the center ring so that they each run over the pulley
and attach to a mass holder as shown in the figure below. Hang the
appropriate masses (numerically equal to the forces in grams) on each string.
Is the ring in equilibrium? Set up a fourth pulley and mass holder at 180
degrees opposite from the angle you calculated for the resultant of the first
three vectors. Record all mass and angles. If you now place a mass on this
fourth holder equal to the magnitude of the resultant, what happens? THE RING WAS IN EQUILIBRIUM right in the middle not touching the center point. Ask
your instructor to check your results before going on.

                                                            force table set up
4. Confirm your results via simulation:
http://phet.colorado.edu/en/simulation/vector-addition
Add the vectors and obtain a resultant.

                                              results are verified thru simulation
Errors:human error in correctly lining up the masses with their respected angles on the table and also error in the  vector addition calculations.the table not being properly leveled.
conclusion: we were able to learn how to study vector addition by both graphically and by components, we found out that the results are more accurate when using vector components you just have to be careful that the math is done right or else u wont get equilibrium when the results are checked on a force table.

                    

Hooke's Law and the simple motion of a spring

Purpose: To determine the force constant of a spring and to study the motion of a spring and mass when vibrating under influence of gravity.

Equipment: Spring, masses, weight hanger, meter stick, support stand with clamps, motion detector, LabPro interface, wire basket.

Introduction: When a spring is stretched a distance x from its equilibrium position, it will exert a restoring force directly proportional to this distance. We write this restoring force, F, as:
1) F = -kx
where k is the spring constant and depends on the stiffness of the spring. The minus sign remind us that the direction of the force is opposite to the displacement. Equation 1 is valid for most springs and is called Hooke’s Law.
If a mass is attached to a spring that is hung vertically, and the mass is pulled down and released, the spring and the mass will oscillate about the original point of equilibrium. Using Newton’s second law and some calculus we can show that the motion is periodic (repeats itself over and over) and has period, T (in sec), given by
2) T = square root of [(4 pi ^2)/k] *m

Procedure:
1. Hang the spring on the support rod, as shown in the diagram, and measure the position of the lower end of the spring. Place 350 gm mass on the spring and observe its position. Now attach, in turn, masses of 450, 550, 650, 750, 850, 950, 1050 gm and measure how far the spring is stretched for each of these masses

                                                                 set up and measurements
                        top end is 1.244m                                                    bottom end is 1.051m

2. Make a plot of the downward force applied to the spring (y-axis) versus the displacement of the spring (x-axis). Remembering Equation 1, determine the force constant, k.
3. Start up the Logger Pro software by clicking on its icon in the Physics Apps folder. From the program click on Open/Mechanics/Hooke’s Law to open the file for this experiment. A graph of linear position vs time should appear. Place the motion detector on the floor beneath the hanging mass. Place the wire basket over the motion detector to protect it from any accidentally dropped masses. Once again place the 350 gm mass on the spring and pull the mass downward until the spring has been stretched 10 cm. Release the mass and observe the subsequent motion. Start collecting data by clicking on the Collect button. The time scale on your graph should allow for at least five cycles of the motion to be seen. Press the “x=” button and determine the time for five cycles. From this number calculate the period of motion. Record your data. Repeat this for the other masses used in part 1. Create a data table which gives average T and m values.

                                                                       data table
4. Using the data for the last trial (with m = 1050 g), fit the data to a sinusoidal function using the Analyze/Curve Fit option. Determine the period and the amplitude from your function. Compare the period with the value obtained in part 3.

                                                sinusoidal function using the Analyze/Curve Fit
                                                        .0615 amplitude in last trial
5. Make a plot of T2 (y-axis) vs m (x-axis). What conclusions can you reach about the validity of equation 2. From this equation, what should the slope of the line be? positive slope Find the slope from your graph and use it to calculate a value for k. Compare with the value of k obtained in part 1.

                                                                     m is 1.67 ( slope)
errors and conclusions: sources of errors are measuring and human movements as we take motion readings.
we were able to find the force constant of a spring
F= -kx so the constant of a spring is -k = F/x
and was verified by the results of the experiment.

Inelastic Collisions

Purpose:
To analyze the motion of two low friction carts during an inelastic collision and verify that the law of
conservation of linear momentum is obeyed.

Equipment:
Computer with Logger Pro software, lab pro, motion detector, horizontal track, two carts, 500 g
masses(3), triple beam balance, bubble level

Introduction:
This experiment uses the carts and track as shown in the figure. If we regard the system of the two
carts as an isolated system, the momentum of this system will be conserved. If the two carts have
a perfectly inelastic collision, that is, stick together after the collision, the law of conservation of
momentum says
Pi = Pf
m1v1 + m2v2 = (m1 + m2)V
where v1 and v2 are the velocities before the collision and V is the velocity of the combined mass
after the collision.

mass of cart 1 is 506g mass of cart 2 is 499.5g
Procedure:
1. Set up the apparatus as shown in Figure 1. Use the bubble level to verify that the track is as
level as possible. Record the mass of each cart. Connect the lab pro to the computer and the
motion detector to the lab pro. On the computer, start the Logger Pro software, open the
Mechanics folder and the Graphlab file.

2. First, check to see that the motion detector is working properly by clicking the Collect button to
start collecting data. Move the cart nearest the detector back and forth a few times while
observing the position vs time graph being drawn by the computer. Does it provide a
reasonable graph of the motion of the cart? Remember to be aware of unwanted reflections
caused by objects in between the motion detector and the cart. Also, position the carts so that
their velcro pads are facing each other. This will insure that they will stick together after the
collision.
TABLE
carts
computer
motion
detector
track m1 m2
v1
3. With the second cart (m2) at rest give the first cart (m1) a moderate push away from the motion
detector and towards m2. Observe the position vs time graph before and after the collision.
What should these graphs look like? Draw an example:

                                                              sketch of graphs
The slope of the position vs. time graph directly before and directly after the collision give the
velocity directly before and directly after the collision. To avoid the problem of dealing with
friction forces (Remember, we are assuming the system is isolated.), we will find the velocity of
the carts at the instant before and after the collision.
Is this a good approximation? Why or why not?
No, the percent diff is too large, not good approx


For the velocity before the collision, select a very small range of data points just before the
collision. Avoid the portion of the curve which represents the collision. Choose
Analyze/Linear Fit. Record the slope (velocity) of this line. Repeat for a very small range of
data points just after the collision. Record this slope (velocity) as well.

                                               analyze

4. Repeat for two more collisions. Calculate the momentum of the system the instant before and
after the collision for each trial and find the percent difference. Put your results in an Excel data
table. Show sample calculations here:
5. Place an extra 500 g on the second cart and repeat steps 3 and 4. Sketch one representative
graph showing the position vs time for a typical collision. (What do velocity vs. time and
acceleration vs. time look like?
6. Remove the 500 g from the second cart and place it on the first cart. Repeat steps 3 and 4.

                                                   calculations and data for steps 4,5,6
7. Find the average of all of the percent differences found above. This average represents your
verification of the law of conservation of linear momentum. How well is the law obeyed based
on the results of your experiment? Explain.
the average percent diff was 8.9% the higher the weight the better the % diff so the law is obeyed well, sources of error account for the diff in %diff

8. For each of the nine trials above calculate the kinetic energy of the system before and after the
collision. Find the percent kinetic energy lost during each collision. Put this information in a
separate data table. Show sample calculations here:

                                               calculations for the % kinetic energy lost
9. Do a theoretical calculation for ΔK/K in a perfectly inelastic collision for the three situations:
1. a mass, m, colliding with an identical mass, m, initially at rest.
2. a mass, 2m, colliding with a mass, m, initially at rest.
3. a mass, m, colliding with a mass, 2m, initially at rest.

                                  theoretical calculation for delta K/K in perfect inelastic collision

Conclusions and errors
• If momentum is conserved, is kinetic energy also conserved? momentum is definitely conserved but kinetic energy is not it is lost as heat.
errors: people moving in front of motion detector and human error in calculations and the track not being perfectly leveled.

Balanced Torques and center of Gravity

Purpose: To investigate the conditions for rotational equilibrium of a rigid bar and to determine the center of gravity of a system of masses.

Equipment: Meter stick, meter stick clamps (knife edge clamp), balance support, mass set, weight hangers, unknown masses, balance.

Introduction: The condition for rotational equilibrium is that the net torque on an object about some point in the body, O, is zero. Remember that the torque is defined as the force times the lever arm of the force with respect to the chosen point O. The lever arm is the perpendicular distance from O to the line of action of the force.
m1  
m2
m3
x1
x3
x2
O

Procedure:
Note: In each of the following steps, where appropriate, make a careful sketch showing the meter stick with the applied forces and mark their locations. Also, show the point, O, about which you are calculating torques.
1. Balance the meter stick in the knife edge clamp and record the position of the balance point. What point in the meter stick does this correspond to? 49.6cm
2. Select two different masses (100 grams or more each) and using the meter stick clamps and weight hangers, suspend one on each side of the meter stick support at different distances from the support. Adjust the positions so the system is balanced. Record the masses and positions. Is it necessary to include the mass of the clamps in your calculations?yes because they are part of our total masses  Sum the torques about your pivot point O and compare with the expected value.
3. Place the same two masses used above at different locations on the same side of the support and balance the system with a third mass on the opposite side. Record all three masses and positions. Calculate the net torque on this system about the point support and compare with the expected value.

                                                   work done for steps 2 and 3
4. Replace one of the above masses with an unknown mass. Readjust the positions of the masses until equilibrium is achieved, recording all values. Using the equilibrium condition for rotational motion, calculate the unknown mass. Measure the mass of the unknown on a balance and compare the two masses by finding the percent difference.
5. Place about 200 grams at 90 cm on the meter stick and balance the system by changing the balance point of the meter stick. From this information, calculate the mass of the meter stick. Compare this with the meter stick mass obtained from the balance. Should the clamp holding the meter stick be included as part of the mass of the meter stick? yes because it is part of the same system of masses and the results will be inacurate.

                                                       work done on steps 4 and 5

6. With the 200 grams still at the 90 cm mark, imagine that you now position an additional 100 grams mass at the 30 cm mark on the meter stick. Calculate the position of the center of gravity of this combination (two masses and meter stick). Where should the point of support on the meter stick be to balance this system? Check your result by actually placing the 100 g at the 30 cm mark and balancing this system. Compare the calculated and experimental results.

                                                          work done on step 6
Errors: human measuring and not being perfectly leveled, and it is not an 90 degree angle
conclusion: we were able to investigate the conditions for rotational equilibrium of a rigid bar and to determine the center of gravity of this system of masses, by balancing the torques of the masses, torque is tendency of force to rotate an object. its magnitude depends on force, length and angle.
Torque = Force X lever arm
= F ( length Sin angle)

Human Power

Purpose: To determine the power output of a person

Equipment: two meter metersticks, stopwatch, kilogram bathroom scale

Introduction: Power is defined to be the rate at which work is done or equivalently, the rate at which energy is converted from one form to another. In this experiment you will do some work by climbing from the first floor of the science building to the second floor. By measuring the vertical height climbed and knowing your mass, the change in your gravitational potential energy can be found:
Δ PE = mgh
Where m is the mass, g the acceleration of gravity, and h is the vertical height gained. Your power output can be determined by
Power =
Δ PE ,
Δt
where Δt is the time to climb the vertical height h.

Procedure:
1. Determine your mass by weighing on the kilogram bathroom scale. Record your mass in kg..

my mass 874N

                                               classmates waiting to get their mass in N

2. Measure the vertical distance between the ground floor and the second floor for the science building. This can most easily be done by using two meter long metersticks held end to end in the stairwell at the west end of the building. Make a careful sketch of the stairwell area that explains the method used to determine this height.
height 2.014m   whole height 4.26m
3. Designate a record keeper and a timer for the class. At the command of the timing person, run or walk (whatever you feel comfortable doing) up the stairs from the ground floor to the second floor. Be sure that your name and time are recorded by the record keeper.
my time 4.63 sec
4. After everyone in the class has completed one trip up the stairs, repeat for one more trial.
2nd trial 4.50 sec
5. Return to class and calculate your personal power output in watts using the data collected from each of your climbing trip up the stairs. Obtain the average power output from the two trials.
874N x 4.29m / 4.565 = 821.35W
6. Put your average power on the board and then calculate the average power for the entire class once everyone has reported their numbers on the board.
class avg 634.97W class avg 0.820504 HP
7. Determine your average power output in units of horsepower. 821.35W is 1.1HP

conclusion, Errors and Questions: 1. Is it okay to use your hands and arms on the handrailing to assist you in your climb up the stairs? Explain why or why not.
no its not cause that will give us a normal force upwards, decreasing the grounds normal force, weight will be lower and power will be higher. results will not be accurate.
2. Discuss some of the problems with the accuracy of this experiment.
putting hands on rails and human error with measuring height and time.
 we learned determine the power output of a person and how gravitational potential energy is proportionate to gravitational force, mass and height and how Power is proportional to gravitational potential energy over change in time.

                                                              Followup Questions

                                                                           answers!

Centripital Force

Purpose: To verify Newton’s second law of motion for the case of uniform circular
motion.

Equipment: Centripetal force apparatus, metric scale, vernier caliper, stop watch,
slotted weight set, weight hanger, triple beam balance.

Introduction:
The centripetal force apparatus is designed to rotate a known mass trough
a circular path of known radius. By timing the motion for a definite
number of revolutions and knowing the total distance that the mass has
traveled, the velocity can be calculated. Thus the centripetal force, F,
necessary to cause the mass to follow its circular path can be
determined from Newton’s second law.
F=mv^2/r
Where m is the mass, v is the velocity, and r is the radius of the circular
path.
Here we have used the fact that for uniform circular motion, the
acceleration, a, is given by:
a= V^2/r

Procedure:
1. For each trial the position of the horizontal crossarm and the vertical indicator
post must be such that the mass hangs freely over the post when the spring is
detached. After making this adjustment, connect the spring to the mass and
practice aligning the bottom of the hanging mass with the indicator post while
rotating the assembly.
2. Measure the time for 50 revolutions of the apparatus. Keep the velocity as
constant as possible by keeping the pointer on the bottom of the mass aligned with
the indicator post. A white sheet of paper placed as a background behind the
apparatus can be helpful in getting the alignment as close as possible. Using the
same mass and radius, measure the time for three different trials. Record all data
in a neat excel table (see 6).

                                          lab setup during trials, very concentrated individuals here

3. Using the average time obtained above, calculate the velocity of the mass. From
this calculate the centripetal force exerted on the mass during its motion.

4. Independently determine the centripetal force by attaching a hanging weight to
the mass until it once again is positioned over the indicator post (this time at rest).
Since the spring is being stretched by the same amount as when the apparatus was
rotating, the force stretching the spring should be the same in each case.
a. Calculate this force and compare with the centripetal force obtained
in part 3 by finding the percent difference.
b. Draw a force diagram for the hanging weight and draw a force
diagram for the spring attached to the hanging mass:

5. Add 100 g to the mass and repeat steps 2, 3 and 4 above.
6. The following data should be calculated and recorded in your excel table:
a. Mass and radius for each trial.
b. Average number of revolutions/sec (frequency) for each trial.
c. Linear speed for each trial.
d. Calculated and measured centripetal force for each trial and their percent
difference.

                                            excel spreadsheet with all trials
                            first 5 trails   m=475g r=16.5cm             v=2pi*r*f     a=v^2/r
                             last 5 trails   m=575g r=16.5cm             f=50/t           f=mv^2/r
the percent diff for first 5 trails
1. 0.48% 2. 0.48% 3. 0.48% 4. 12.25% 5. 8.26% AVG 4.44%
last 5 trails
1. 12.52% 2. 1.08% 3.3.83% 4. 2.45% 5. 0.44% AVG 3.83%
Errors: human error in using the stopwatch and keeping the velocity as constant as positive also theres air resistance.
conclusion:
we were able to verify newtons second law of motion for uniform circular motion cause acceleration is proportional to the net force acting on it. thru our results and calculations and can make the assumption that with a constant radius, greater mass the lower the centripetal force is.

Drag Forces on a Coffee Filter

Purpose: To study the relationship between air drag forces and the velocity of a falling body.

Equipment: Computer with Logger Pro software, lab pro, motion detector, nine coffee filters, meter stick

 FD = k |v|n, where the power n is to be determined by the experiment.

Procedure:
NOTE: You will be given a packet of nine nested coffee filters. It is important that the shape of this
packet stays the same throughout the experiment so do not take the filters apart or otherwise
alter the shape of the packet. Why is it important for the shape to stay the same?BECAUSE IF THE SHAPE CHANGES SO WILL THE DRAG.
1. Login to your computer with username and password. Start the Logger Pro software, open the
Mechanics folder and the graphlab file. Don’t forget to label the axes of the graph and create an
appropriate title for the graph. Set the data collection rate to 30 Hz.
2. Place the motion detector on the floor facing upward and hold the packet of nine filters at a minimum
height of 1.5 m directly above the motion detector. (Be aware other of nearby objects which can
cause reflections.) Start the computer collecting data, and then release the packet. What should the
position vs time graph look like?

 the graph should look like a line with a negative slope

Verify that the data are consistent. If not, repeat the trial. Examine the graph and using the mouse,
select (click and drag) a small range of data points near the end of the motion where the packet
moved with constant speed. Exclude any early or late points where the motion is not uniform.
3. Use the curve fitting option from the analysis menu to fit a linear curve (y = mx + b) to the selected
data. Record the slope (m) of the curve from this fit. What should this slope represent? TERMINAL VELOCITY

                                                    position vs time graph of the filters with curve analysis
.
Repeat this measurement at least four more times, and calculate the average velocity. Record all
data in an excel data table.

4. Carefully remove one filter from the packet and repeat the procedure in parts 2 and 3 for the
remaining packet of eight filters. Keep removing filters one at a time and repeating the above steps
until you finish with a single coffee filter. Print a copy of one of your best x vs t graphs that show the
motion and the linear curve fit to the data for everyone in your group (Do not include the data table;
graph only please).
5. In Graphical Analysis, create a two column data table with packet weight (number of filters) in one
column and average terminal speed (|v|) in the other. Make a plot of packet weight (y-axis) vs.
terminal speed not velocity (x-axis). Choose appropriate labels and scales for the axes of your
graph. Be sure to remove the “connecting lines” from the plot. Perform a power law fit of the data
and record the power, n, given by the computer. Obtain a printout of your graph for each member of
your group. (Check the % error between your experimentally determined n and the theoretical
value before you make a printout – you may need to repeat trials if the error is too large.)

                                                        excel spreadsheet of data table

                                                      average terminal velocity curve
6. Since the drag force is equal to the packet weight, we have found the dependence of drag force on
speed. Write equation 1 above with the value of n obtained from your experiment. Put a box around
this equation. Look in the section on drag forces in your text and write down the equation given there
for the drag force on an object moving through a fluid. How does your value of n compare with the
value given in the text? our value of n is close to 2, matching n=2 therefore it is parabolic and our calculations were accurate.
Errors: cross sectional area, messing up shape of filters, air coming in and pushing the filter away. human interference with the motion detector.
conclusion:we were able to learn about the relationship between air drag forces and velocity of a falling object, when speed increases drag increases, when acceleration decreases drag increases.