The Ballistic Pendulum

Purpose: To use the ballistic pendulum to determine the initial velocity of a projectile using conservation of momentum and conservation of energy.

Equipment: Ballistic pendulum, carbon paper, meter stick, clamp box, triple beam balance, plumb.

Introduction: In this experiment a steel ball will be shot into the bob of a pendulum and the height, h, to which the pendulum bob moves, as shown in Figure 1, will determine the initial velocity, V, of the bob after it receives the moving ball.
                

Figure 1

Figure 2

                  = (194+56.8) ×√(2×9.8×0.085) / 56.8                △y = voyt + 1/2ayt^2

If we equate the kinetic energy of the bob and  ball at the bottom to the potential energy of the bob and ball at the height, h, that they are raised to, we get:

                                                       (K.E)bottom = (P.E)top
                                                       1/2( M+m) V^2 = ( M+m) gh

      Where M is the mass of the pendulum and m is the mass of the ball. Solving for V we get:

                                                        V = √(2gh) ----------(1)

Using conservation of momentum we know the momentum before impact (collision) should be the same as the momentum after impact. Therefore:
                                                                      
                                                                   pf = pi 
                                                     or
                                                              mvo= (M+m)V -----------(2)

       Where vo is the initial velocity of the ball before impact. By using equations (1) and (2) we can therefore find the initial velocity, vo, of the ball.
       We can also determine the initial velocity f the ball by shooting the ball as above but this time allowing the ball to miss the pendulum bob and travel horizontally under the influence of gravity. In this case we simply have a projectile problem where we cam measure the distance traveled horizontally and vertically (see Figure 2) and then determine the initial velocity, vo, of the ball.

Starting with equations:
          
                                                       △x = voxt + 1/2axt^2 -------------(3)
                                                       △y = voyt + 1/2ayt^2 -------------(4)

       You should be able to derive the initial velocity of the ball in the horizontal direction (assuming that and known).

Procedure:

Part I Determination of Initial Velocity from Conservation of Energy

1. Set the apparatus near one edge of the table as shown in figure 2. Make sure that the base is accurately

horizontal, as shown by a level. Clamp the frame to the table.

To make the gun ready for shooting, rest the pendulum on the rack, put the ball in position on the end of the

rod and, holding the base with one hand, pull back on the ball with the other until the collar on the rod

engages the trigger. This compresses the spring a definite amount, and the ball is given the same initial

velocity every time the gun is shot.

2. Release the pendulum from the rack and allow it to hang freely. When the pendulum is at rest, pull the

trigger, thereby propelling the ball into the pendulum bob with a definite velocity. This causes the pendulum

to swing from a vertical position to an inclined position with the pawl engaged in some particular tooth of the

rack.

3. Shoot the ball into the cylinder about nine times, recording each point on the rack at which the pendulum

comes to rest. This in general will not be exactly the same for all cases but may vary by several teeth of the

rack. The mean of these observations gives the mean highest position of the pendulum. Raise the pendulum

until its pawl is engaged in the tooth corresponding most closely to the mean value and measure h1, the

elevation above the surface of the base of the index point for the center of gravity. Next release the pendulum

and allow it to hang in its lower most position and measure h2. The difference between these two values gives

h, the vertical distance through which the center of gravity of the system is raised after shooting the ball.

Record h:

4. Carefully remove the pendulum from its support. Weigh and record the masses of the pendulum and of the    ball. Replace the pendulum and carefully adjust the thumb screw.

M (mass of pendulum) = 194g
m ( mass of the ball) = 56.8g

5. From these data calculate the initial velocity v using equations (1) and (2).

               V = √(2gh) 
               mvo= (M+m)V

     

               vo= (M+m) ×√(2gh) / m

                        = 5.7 m/s

Part II: Determination of initial velocity from measurements of range and fall

1. To obtain the data for this part of the experiment the pendulum is positioned up on the rack so that it will not interfere with the free flight of the ball. One observer should watch carefully to determine the point at which the ball strikes the floor. The measurements in this part of the experiment are made with reference to this point and the point of departure of the ball. Clamp the frame to the table. as it is important that the apparatus not be moved until the measurements have been completed. A piece of paper taped to the floor at the proper place and cover with carbon paper will help in the exact determination of the spot at which the ball strikes the floor.

2. Shoot the ball a number of times, nothing each time the point at which it strikes the floor. Determine, relative to the edge of the paper, the average position of impact of the ball. Determine the distances △x and △y calculate vo by the use of equations (3) and (4). Make careful stretches in your lab report show all of the  distances involved.

      The distance from the ball to the paper: 258.4cm
      The distance of the ball on the paper: 17.4cm

  △x = 258.4+17.6= 276cm
       △y(height) = 99.7cm

       0.997 = 0 + (1/2) × 9.8 × t^2 
               t = 0.45s

          △x = voxt + 1/2axt^2
        2.76 = vo × 0.45 +0 
           vo = 6.1 m/s

     
      Percent of difference between part I and part II:
      
      (6.1-5.7) / [(6.1+5.7) /2] = 6.8%

3. Find the percentage difference between the values of v0 determined by the two methods in parts I and II. Try

to analyze, the probable errors of the two methods and estimate which one should give the more accurate

result.

- The one with out the pendulum because it loses energy when it hits the pendulum whereas the one without the pendulum is only affected by gravity.

Conclusions:

This lab has taught me about conservation of momentum and energy. This is true if there is no external forces acting upon the system. The equations gave us the necessary tools to calculate the appropriate values. Some sources of error can be the table. The table, we found out, is not exactly level so that could have thrown our calculations off. We also neglected air resistance and friction which may affect the final calculations. 

Human Power

Purpose: To determine the power output of a person
Equipment: two meter meter sticks, stopwatch, kilogram bathroom scale
Introduction: This lab includes an experiment involved that allows people to run or walk up stairs and get timed. After they were timed, the person would come back and calculate the horsepower that they outputted. The way to do this is by measuring the vertical height climbed, and knowing your mass, the change in potential energy can be found. This is given by the equation:

(change in PE) = mgh
where m is the mass, g is the acceleration of gravity, and h is the vertical height gained. Power output can be determined by the equation:

Power = (change in PE) / (change in time)
where change in time is the time it takes to climb the vertical height.

Procedure: 

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

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.
4. After everyone in the class has completed one trip up the stairs, repeat for one more trial.
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.
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.
7. Determine your average power output in units of horsepower.

Data:

 h: 4.29m
mg: 855N

△t = 4.10s

Power = mgh / △t = 855N × 4.29m / 4.10s = 864.62 J/s =864.62 W = 1.1997 HP

Questions:

1. Is it okay to use your hands and arms on the hand railing to assist you in your climb up the stairs? Explain why or why not.
- No because horse power is mainly described by force from the legs and when you are being assisted by the hand rails, you are pulling yourself up, consequently your legs are working less than normal and it is not calculating true horsepower.

2. Discuss some of the problems with the accuracy of this experiment.
- Some errors could have been the time keepers watch could have been off by a couple of milliseconds, depending on the reactions of the timekeeper. Also, another source of error could be the measurement of the height of the staircase could be off by a couple of centimeters.

Follow up questions:
1. Two people of the same mass climb the same flight of stairs. Hinrik climbs the stairs in 25 seconds. Valdis takes 35 seconds. Which person does the most work? Which person expands the most power? Explain your answers.
- They both did the same amount of work but Hinrik expanded the most power because it took him a shorter time to climb the same height.


2. A box that weights 1000 Newtons is lifted a distance of 20.0 meters straight up by a rope and pulley system. The work is done in 10.0 seconds. What is the power developed in watts and kilowatts.
- (1000N*20) / 10s = 2000 watts = 2 kilowatts

3. Brynhildur climbs up a ladder to a height of 5.0 meters. If she is 64 kg:
 a) What work dose she do?
64*(9.8)^2 * 5 = 3136J

b) What is the increase in the gravitational potential energy of the person at this height?
-It would have a 3136J change because PE and work are found by the same equation.

c) Where does the energy come from to cause this increase in P.E.?
-You must use kinetic energy to climb the height of the stairs and the more you climb, the more PE you gain.

4. Which requires more work: lifting a 50 kg box vertically for distance of 2m , or lifting a 25kg box vertically for a distance of 4 meters?
-They require the same amount.
25 * 4 * 9.8^2
50 * 2 * 9.8^2

Conclusion:
 This lab was really fun and educational. It taught us the definition of power and potential energy. By measuring height, mass, and time, we were able to determine horse power. Some source of error was the height measurement, time, and as a result, the power would not be accurate. 

Centripetal Force

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

Introduction: The apparatus that we used in the experiment is the centripetal force apparatus.

A pictoral of what the apparatus looks like

The centripetal force apparatus works so that we can find the centripetal force, F.  By timing the motion for fifty revolutions, and knowing the total distance that the mass has traveled, we can find the velocity. We also already know what the radius is from the apparatus, so therefore, the following equation determines, from Newton’s second law, how much force is necessary to cause the mass to follow its circular path.

The centripetal force equation

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 fifty (50) revolutions of the apparatus. Keep the velocity as constant as possible. Use the same mass and radius, measure the time for five different trials. Record the data in a Microsoft Excel sheet.

3.       Using the average time obtained in procedure (2), 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 is positioned over the indicator post (this time at rest).

a.       Calculate this force and compare with the centripetal force obtained in procedure (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 grams to the mass and repeat procedures 2, 3, and 4.

Data:

Our data tables

Calculation:

Calculation when m = .475, v = 1.57, and r = .165

Conclusion:

In the lab, I learned how to obviously find the centripetal force of a mass that follows a circular pattern, which accomplished the purpose of the lab. Also, I learned how to operate a centripetal force apparatus. Although we did not experience a perfect lab, we did brainstorm what could have been sources of error. One thought is that the timer could have started the stopwatch either early or late (or both for that matter). Another thought we had was that the radius could have been off by a couple centimeters. Finally we thought that the calculations can be off because the velocity may have not been constant. This leads me to a way we can improve this experiment. One way is to put a machine that delivers constant energy that moves the bob at a constant speed. Overall, this experiment was very educational and enjoyable at the same time.

Drag Force on a Coffee Filter

Intro:     The purpose of this experiment was to study the relationship between air drag forces and the velocity of a falling body. What we did in this experiment was we started off with nine coffee filters. When we dropped the pack of nine filters over a motion detector, it registered how fast the pack got to the motion detector. After four or five trials, we subtracted a filter and did the four or five trials with the eight filters. We continued to subtract a filter after four or five trials until we had no more filters left. In the end, we analyzed our data and placed it into a graph. The data told us what the terminal speed relative to the number of filters and also, how close to the Power Law fit our numbers were.

Questions: Some questions that were incorporated into the procedure tested our analysis of the procedure.

1) What should the position vs. time graph look like? Explain.   

<!--[endif]--> <!--[if !vml]-->
  -      I thought the position vs. time graph should look like this because as time elapses, the filters   will get closer to the motion detector. 

2)    From the curve fitting and analysis graph, what should the slope represent? Explain

<!--[if !supportLists]-->-          <!--[endif]-->The slope should equal the terminal velocity of the falling coffee filters.

Results: 

The data table from all our trials. The average terminal velocity is shown on the right side.

This graph shows a representation of what the slope for on of the 8 filters experiment

Conclusion: The actual value of A*x^B is B should equal 2. Our value was 2.55. Therefore, our percentage error was 27.5%. This was a very large error and we contribute the "off-ness" to the shape of the filters. The instructions clearly said to keep the filters straight and we tried, but it was hard when we dropped the last filter more than 50 times. Also, we tried to get the best fit on the slope and the fit was sometimes off by a lot. In the end, I learned about mass and how it affects the velocity of a falling object and also how the air drag force is related to it.

The red points show our data while the solid line shows what the true value of A*x^B is equal to.

Working with Spreadsheets

Purpose/Introduction: The purpose of this experiment was to get familiar with electronic spreadsheets (i.e. Microsoft Excel) by using them in some simple applications. Some applications used were adding, multiplying, and dividing cells and displaying them in a separate cell. The first step to the experiment was to open a Microsoft Excel spreadsheet that calculated the function,

                f(x) = Asin(Bx) + C

                whereas:                   A = 5

                                                B = 3

                                and          C = π/3

Procedure: The procedure was to place these three variables, A, B, and C, on the right side of the spreadsheet and consider them constants. Then, we were to place a column for “x” and a separate column for “f(x).” The “x” column was to start at 0 radians and continue to 10 radians, in increments of .1 radians. The trick was to NOT enter the x values by hand, rather, use the copy feature to complete this action. After completing this task, we did the same but with the f(x) values and that was it for Step 1 of the spreadsheet part of the lab.

Next we placed the data into Graphical Analysis and received a sine graph. We fitted our curve to the same equation and our results were as follows:

A = 5.00

B = 3.00

C = 1.05

We can say we expected these results because the equation from the spreadsheet was exactly the equation from the Graphical Analysis, therefore, for every x value, the f(x) value was expected to be the same in both programs.

After this part of the lab, we opened another spreadsheet and did the exact same thing with a different equation with different constant variables. The equation was:

Whereas:            initial velocity = 50 m/s

                                Initial position = 1000 m

                                Change of time = 0.2 s

                                And g = -9.8m/s^2

Again, we were to input x values from 0 to 10, but this time in increments of .2 and not by hand. We continued by using the copy feature to complete both the x and f(x) values.

Data:

When we received our f(x) values, we copied and pasted them into Graphical Analysis; consequently, we received a concave down parabolic equation. We fitted our curve to the equation, y = Ax^2 + Bx + C and our results were as follows:

A = -4.90

B = 50.0

C = 1000

We can say we expected these results because the equation from the spreadsheet was exactly the equation from the Graphical Analysis, therefore, for every x value, the f(x) value was expected to be the same in both programs. Also, A is equal to exactly half of the acceleration of gravity. B is equal to the initial velocity and C is equal to the initial position.

Conclusion: Some of the graphs were off, but after re-fitting the equations, the Graphical Analysis program gave us the desired values. The results that the program gave us we some absurd value that could not be true; therefore, we knew we had to re fit the curve. Also, there was some confusion on the second part with the acceleration of gravity, but if you read the problem, the object must be going up and then falling. 

Vector Addition of Forces

Purpose/Introduction: The purpose of this lab was to study vector addition using graphs and also using components. We also used a circular table to confirm our results. To begin, we were given three different vector magnitudes and their angles. The vectors that were given to us are represented in this table.

Vector	Magnitude (grams)	Angle (Degrees)
A	200	0
B	100	55
C	200	135

We decided that instead of drawing a vector with a magnitude of 200 cm, we would scale the magnitudes to 1 cm = 50 g. Therefore, the magnitudes to draw were 4 cm (A), 2 cm (B), and 4 cm (C). The next procedure was to draw a vector diagram of the three vectors and find the resultant vector using a protractor and straight edge.

Vector diagram of vectors A, B, and C. Vector D is A + B + C.

The resultant force from the vector diagram was D. The magnitude of D was approximately 4.8 centimeters or about 240 grams. This means that in order to balance the circular force table, you would need to put approximately 240 grams on the fourth pulley.

The second part of the lab was to find the resultant vector (D) by using components. We graphed the vectors according to the scale and the angle given. We used sine and cosine to determine the opposite and adjacent sides. 

Data:
î represents the x coordinate

ĵ represents the y coordinate

cosθ= î /hypotenuse

sinθ= ĵ / hypotenuse

Vector	î substituted	î	ĵ substituted	ĵ
A	4cos0 = î	4	4sin0 = ĵ	0
B	2cos55 = î	1.147	2sin55 = ĵ	1.638
C	4cos45 = î	-2.828	4sin45 = ĵ	2.828

We calculated Vector D to have a magnitude of 251 grams at 62.5°. The way we found that out is that we added the three vectors and received two components, î and ĵ. We took the arctangent of (ĵ/ î) and received the angle of Vector D. In order to find the magnitude of D, we added î² and ĵ² and took the square root of the sum. We received an answer of 251, which was consequently our magnitude.

The next step was to confirm our results by setting up a force table and arranging the pulleys so that they represent our given vectors at the given angles. After placing the three pulleys on the table, the ring in the center was NOT at equilibrium. We placed a fourth pulley at 242.5° (180° + 62.5°) with 250 grams and the ring equaled out.

Circular force table that shows Vectors A, B, C, and D (resultant vector)

The next and final step was to confirm our results using a university website. We confirmed that the sum of our vectors was approximately 252 grams and about 62.5°.

Vector program from http://phet.colorado.edu/en/simulation/vector-addition

Conclusion: This lab showed that vectors can be calculated in many ways. Today’s lab demonstrated two ways to calculate the sum of more than one vector. The first way is by a graph. Although this method is less accurate than components, it is still effective. The reason why some calculations are off is because the measuring devices (rulers, protractors, ect.) are not always accurate and one is basing the measurement of the resultant on the measuring device. This is one source of error of the graphing method. The other method has a more accurate measurement. The component method uses trigonometry to calculate the angle and magnitude of the resultant vector. This is far more accurate than the graphical method because one is using angles and magnitudes rather than a ruler to measure a vector. There is hardly any error with this method. With the website that we were supposed to check our resultant vector, the reason that was off is because the program only allowed certain magnitudes and angles; consequently, one is not able to put in the correct magnitudes and angles or the given vectors. Using the component method will ultimately give one the most accurate resultant vector.

Acceleration of Gravity on an Inclined Plane

Introduction: The purpose of this lab was to study the acceleration of gravity by studying the motion of a cart on an incline. Also, the gain further experience using the computer for data collection and analysis. In this lab, we used the computer to collect position vs. time data for a cart accelerating on an inclined track. The effect of friction was eliminated by comparing the acceleration of the cart when moving up and down the track. We averaged the slightly increased acceleration (when going up) with the decreased acceleration (going down) to obtain the acceleration that depends only on force of gravity. If we let g be the acceleration when the cart moves up and down the track, then we still have to consider the angle of the track. Therefore, we can use 

where the a’s are the accelerations of the up and down incline. We measured the acceleration by looking at the slope of the v vs. t curve for the cart.

Procedures: The first step to this lab was to assemble the track and measure the angle of the incline. The way our group did it was that we took side A and subtracted side B and then divided it by side C. This gave us the sinθ. After multiplying both sides by acrsin, we got the angle of our incline track. Assuming side C and side D continued until interception, the angle made by these two sides is θ. 

This picture illustrates the above

As we placed the motion detector on the top of the track, we were able to see the velocity vs. time graph. When we pushed the cart up the track, the velocity went from negative to positive. When the velocity was equal to zero, that meant the cart reached its peak and stopped at the top. This video shows the process of pushing the cart up, and then letting it come down the track.

The slope of the interval from the negative to zero velocity was the acceleration as the object was going up the track. On the other hand, the interval from zero to the positive velocity is the when the cart is coming back down the track. The slope represents the acceleration of gravity as the cart is going down the track. 

This graph illustrates one trial at the slope of 1.74° 

Data:

 For the calculations, each trial had the same equation. The equation was as follows:

As for the work, the different trials for both θ is equal to 1.74 and 3.637.

This is the work for the when θ is equal to 1.74

This is the work for the when θ is equal to 3.637

Conclusion: We repeated the experiment with two other angles and it seemed as we increased the angle, the percentage error decreased. Therefore, we concluded that with a steeper incline, a more accurate acceleration of gravity is produced. We also figured that one source of error was because of the accuracy of the meter stick. If we had a stick with smaller measurements, the answer could have been a little closer. Also, the answer could have been thrown off if the table we had the track on was uneven or unbalanced. Although these sources of error could not have been avoided, our data shows that g was off no more that -13%. One thing that can make this experiment better is experimenting with different object with different masses. For example, you can roll a ball up the track; this may or may not affect the results. We learned that gravity is always acting on the cart. The moment when someone pushed the cart, the only force was gravity of the world does not change because you pushed the cart.