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.