Lab – Merrily We Roll Along!

Purpose

To investigate the relationship between distance and time for a ball rolling down an incline.

Required Equipment/Supplies

2-meter ramp

steel ball bearing or marble

wood block

stopwatch

tape

meterstick

protractor

Discussion

In this lab we will be observing the distances that a ball travels down a ramp in successive time intervals.

Procedure

Step 1 (set up ramp):  Set up a ramp with the angle of the incline at about 10º to the table, as shown in Figure A.

Figure A

Step 2 (divide ramp into equal parts):  Divide the ramp’s length into six equal parts and mark the six positions on the board with pieces of tape.  These positions will be your release points.  Suppose your ramp is 200 cm long.  Divide 200 cm by 6 to get 33.33 cm per section.  Mark your release points every 33.33 cm from the bottom.  Place a stopping block at the bottom of the ramp to allow you to hear when the ball reaches the bottom.

Step 3 (time the ball down the incline):  Use a stopwatch to measure the time it takes the ball to roll down the ramp from each of the six points.  Use a ruler or a pencil to hold the ball at its starting position, then pull it away quickly to release the ball uniformly.  Do several practice runs with the help of your partners to minimize error.  Make at least three timings from each position, and record each time and the average of the three times in Data Table A.

 DISTANCE (cm) TIME(S) TRIAL 1 TRIAL 2 TRIAL 3 AVERAGE

Data Table A

Step 4 (graph data):  Graph your data, plotting distance (vertical axis) vs. average time (horizontal axis) on an overhead transparency.  Use the same scales on the coordinate axes as the other groups in your class so that you can compare results.

Step 5 (change the tilt of the ramp and repeat):  Repeat Steps 2 to 4 with the incline set at an angle 5º steeper.  Record your data in Data Table B.  Graph your data as in Step 4.

 DISTANCE (cm) TIME(S) TRIAL 1 TRIAL 2 TRIAL 3 AVERAGE

Data Table B

1.         What is acceleration?

2.         Does the ball accelerate down the ramp?  Cite evidence to defend your

3.         What happens to the acceleration if the angle of the ramp is increased?

Step 6 (reposition tape markers on ramp):  Remove the tape marks and place them at 10 cm, 40 cm, 90 cm, and 160 cm from the stopping block, as in Figure B.  Set the incline of the ramp to be about 10º.

Figure B

Step 7 (time the ball down the ramp):  Measure the time it takes for the ball to roll down the ramp from each of the four release positions.  Make at least three timings from each of the four positions and record each average of the three times in Column 2 of Data Table C.

 COLUMN 1 COLUMN 2 COLUMN 3 COLUMN 4 DISTANCE TRAVELED (cm) ROLLING TIME(S) TIME DIFFERENCES BETWEEN SUCCESSIVE INTERVAL(S) TIME IN “NATURAL UNITS” TRIAL 1 TRIAL 2 TRIAL 3 AVERAGE 10 t4 = 1 40 t2 = t2- t1 = 90 t3 = t3- t2 = 160 t4 = t4- t3 =

DATA TABLE C

Step 8 (graph data):  Graph your data, plotting distance (vertical axis vs. time (horizontal axis) on an overhead transparency.  Use the same coordinate axes as the other groups in your class so you can compare results.

Step 9 (study your data):  Look at the data in Column 2 a little more closely.  Notice that the difference between t2 and t1  is approximately the same as t1 itself.  The difference between t3 and t2 is also nearly the same as t1.  What about the difference between t4 and t3?  Record these three time intervals in Column 3 of Data Table C.

Step 10:  If your values in Column 3 are slightly different from one another, find their average by adding the four values and dividing by 4.  Do as Galileo did in his famous experiments with inclined planes and call this average time interval one “natural” unit of time.  Note that t1 is already listed as 1 “natural” unit in Column 4 of Table C.  Do you see that t2 will equal—more or less—2 units in Column 4?  Record this, and also t3 and t4 in “natural” units, rounded off to the nearest integer.  Column 4 now contains the rolling times as multiples of the “natural” unit of time.

4.         What happens to the speed of the ball as it rolls down the ramp?  Does it

increase, decrease, or remain constant?  What evidence can you cite to

Step 11:  Overlay your transparency graph with other groups in your class and compare them.

5.         Do balls of different mass have different rates of acceleration?

Step 12:  Investigate more carefully the distances traveled by the rolling ball in Table D.  Fill in the blanks of Columns 2 and 3 to see the pattern.

 COLUMN 1 COLUMN 2 COLUMN 3 DISTANCE TRAVELED (cm) FIRST FOUR INTEGERS SQUARES OF FIRST FOUR INTEGERS 10 1 1 40 2 4 90 3 9 160

TABLE D

6.         What is the relation between the distances traveled and the squares of

the first four integers?

Step 13:  You are now about to make a very big discovery—so big, in fact, that Galileo is still famous for making it first!  Compare the distances with the times in the fourth column of Data Table C.  For example, t2 is 2 “natural” time units and the distance rolled in time t2 is 22, or 4, times as great as the distance rolled in t1.

7.         Is the natural distance the ball rolls proportional to the square of the

“natural” unit of time?

The sizes of your experimental errors may help you appreciate Galileo’s

Genius as an experimenter.  Remember, there were no stopwatches 400

Years ago!  Galileo concluded that the distance d is proportional to the

square of the time t.

d ~ t2

Step 14 (increase tilt of ramp):  Repeat Steps 6 to 10 with the incline set at an angle 5º steeper.  Record your data in Data Table E.

8.         What happens to the acceleration of the ball as the angle of the ramp is

increased?

 COLUMN 1 COLUMN 2 COLUMN 3 COLUMN 4 DISTANCE TRAVELED (cm) ROLLING TIME(S) TIME DIFFERENCES BETWEEN SUCCESSIVE INTERVAL(S) TIME IN “NATURAL UNITS” TRIAL 1 TRIAL 2 TRIAL 3 AVERAGE 10 t4 = 1 40 t2 = t2- t1 = 90 t3 = t3- t2 = 160 t4 = t4- t3 =

Data Table E

9.         Instead of releasing the ball along the ramp, suppose you simply dropped

it.  It would fall about 5 meters during the first second.  How far would

it freely fall in 2 seconds?  5 seconds?  10 seconds?