In this lesson, your students will explore how fast they can go on a bicycle. The students begin this lesson by observing the following video clip of a Japanese cyclist on a banked track. The video clips serves as a trigger to introduce the lessons. Move the mouse pointer over the first frame of the clip jbike.avi below to see the video clip. The clip will play twice. Note: The still images are shown at ˝ the size which the students will see in the lesson.
Your next step is to divide the class into small groups. Have them make a list of responses to the question "What must you do to make a bicycle go fast?"
After 10 minutes, go around the room and get a response from each group. Write each response on the board or a flip chart so that all the groups can clearly read the responses as they are given. Tell each group that if one of their answers is already listed, they must give another answer from their list.
In a typical class of three groups, the following are typical answers to the question.
This trigger phase of the lesson has developed some interest in what determines how fast one can make a bicycle go. It serves as a "warm up" or "ice-breaker" for the students and prepares them for the next phase of the lesson that involves some analysis of several video clips.
Goal: Students learn to estimate the work a cyclist can do in one complete revolution of his/her legs on the crank of a bicycle.
In this part of the lesson, you will have the students analyze some video clips.
2. Show the students frame (33166)below which gives mass of Karl, the cyclist, to be (68.1kg) and the mass of the bicycle to be (10.2 kg)
3. Next, show the following frame, which gives the crank length of the bicycle (170 mm).
To get the students started, have them estimate the work done by the cyclist per revolution of his legs. Tell them to assume the rider stands on the pedal at the top and cranks it to the bottom. The change in the potential energy of the rider is converted into work on the bicycle. The work done is the weight of the cyclist times twice the crank length since the crank length represents the radius of the circle through which the pedal moves. Remind them that once they obtain a value, they must multiply this by two since both legs are used in one complete cycle. The work done by Karl is shown below.
Goal: Students gather more information and learn to go from a simple estimation to a more detailed analysis of the problem.
Here you will need the force and the location data from several frames from the energy transformation videodisc of Karl riding a 12 speed sport bicycle with crank length=0.170 m
Ask the students "How would you compute the work done on the cycle by the cyclist?"
Typical student response was to calculate the work done by the x and y components of the force.
Now have the students watch a short animated sequence which shows the x and y force vector components which Karl applies to the pedal with his foot. The animation consists of a set of individual frames digitized from the energy transformation disk. Each frame represents a position of the crank and pedal during one complete revolution of the cyclist’s leg. Data from Professor Hull’s dynamo has been superimposed on each frame. The data shows the Fx and Fy components of force applied to the pedal for that particular position of the pedal. The first three frames of the animation are shown below.
Remember to tell the student: Each frame represents a clockwise advancement of the crank by 30° from the previous frame. Thirteen frames have been combined into a single "animated gif" , a standard format for viewing animations via the WWW.
|
Crank angle |
F x |
F y |
|
0 |
16 |
110 |
|
30 |
77 |
203 |
|
60 |
100 |
250 |
When the students use the lesson in its WWW format, each frame of this animated gif is shown with a four second delay. The entire animation loops repeatedly so that students can double check their data and correct any mistakes in their table.
Ask the question: "What additional information would you need in order to determine how fast Karl can go on his bicycle?" Give the students 10 minutes to generate additional information they would need to know. Some typical questions the students asked were:Watch for key words such as force, number of revolutions, work and power in the students’ answers. These are important flags you can use to guide the students towards determining the energy output of the cyclist. How much energy the cyclist can convert into rotary motion of the wheels in a given amount of time will determine the maximum kinetic energy of the bike/rider obtainable during that time.
Goal: Students learn to use a spreadsheet to analyze data taken from digitized video.
Have the students open a spreadsheet program with the heading shown below and enter the data they recorded in their table.
In column C2, the crank length (0.170) is entered and the units (m) in cell D2.
The fourth column represents the magnitude of the net force applied to the pedal in each crank position. Students who understand how to calculate the magnitude of a vector given its components should not find the formula for cell D4 (=SQRT(B4^2+C4^2) to be difficult.
Next ask the students what they would need to do in order to calculate the x and y displacements of the pedal as the net force moves the crank from one position to another. One approach would be to show them the spreadsheet below and ask them what do the numbers in columns E and F represent. Most students quickly figure out that these are the coordinates of the endpoint of the crank where the force is applied. Once this is established, ask them how would they calculate these numbers?
Hint:
For x, enter into cell E4 the formula = $C$2 *SIN(A4*3.14159/180) and for y, enter into cell F4 the formula =$C$2 *COS(A4*3.14159/180) Then copy these formulas into the lower cells. Note the use of the absolute cell address C2 where the crank length is located. The use of an absolute cell address permits students to enter a different crank length without having to modify the formulae columns E and F.
Now ask the students how they will calculate the work done by Karl from the data above.
Give them some time to talk about how one calculates work. Draw their attention to the force components in the table above and ask them what additional information they would need to calculate the work done as the crank moves from one position to another. After a few minutes you may wish to show Frame 7213 from the Energy Transformations disk if the discussion has not touched on calculating the increments of work done by the x and y components of the force applied to the pedal.
Once students have this idea firmly established, point out to the students that they must calculate the x and y displacements of the crank as it moves from one angle to another as well as the x and y components of the force. Remind them that in this first calculation, the x and y components of force will assume to be constant during the change.
Again draw the students back to the numbers in columns E and F. Ask them if these numbers represent the displacements in x and y in going from one crank position to another. If not, how do you calculate the displacement in x and y? Give students a chance to discuss this. Usually it takes little time for students to realize that D x and D y must be calculated as (xf – xi) and (yf – yi) where xI yi xf and yf are the initial and final x and y coordinates of the end of the crank.
Ask the student to build formulae in their spreadsheet which would calculate these displacements. Where should these formulae be located?
An example of a spreadsheet developed by students to calculate the x and y displacements is shown below. Here, students have entered the following formula into cell G5: =(E5-E4). Note:
This student group has chosen downward to be the positive direction. Hence in cell H5 the following formula has been entered. = -1*(F5-F4)
The students must now build the components of work. Note how one student team has done this in the spreadsheet below. The formula in cell I5 is: =(B4*G5) and for J5 it is: = (C4*H5)
These formulae are copied down through cells I16 and H16. Note the sum of the x and y components of work have been calculated and placed in cells I17 and H17. Finally, the sum of the components have been placed in cell F2.
Other Approaches to the Problem
Another way would be to take an average value of the force in going from one position to another and use this value for the force when multiplying by the displacement between each level. Thus in the spreadsheet above to get the x component of work done in moving the crank from 0 to 30 degrees, you would enter the following formula into cell I5
Visualizing the Forces !
It's true that a picture sometimes is worth a 1000 words. Have the students make graphs of the x and y components of force applied to the pedal versus the crank angle. A graph of the total force vs. crank angle could be made also. Below is a sample of graphs which students have made from data using the average force method.
A third way would be to try to fit a mathematical curve to the force/crank angle data and thus determine Fx and Fy as a functions of theta. Integrating these functions over one revolution would give the work done in one revolution.
Additional Activities:
Go to a fitness room and obtaining real data about a student's power output using exercise machines commonly found in such rooms.
Have a student climb a set of stairs. Knowing the vertical distance climbed and the time to do this would enable one to calculate the power output of the student.
PART 5: ESTIMATING THE POWER OUTPUT OF THE CYCLIST To estimate the power the cyclist delivers to the bike, show the students frame 6752. Here the right crank and foot of the cyclist is almost parallel to the ground. Have the students step through the videoclip and determine the time it takes for the cyclist's foot to return to approximately the same position . They should come up with about 25 frames ending on frame 6777. Since the difference between frames in this clip is 1/30 of a second, they should arrive at a period of 25/30 or about 0.83 seconds.Have them look back at the spreadsheet showing the average force data where the net work for one cycle of one leg was roughly 100 J. Since the cyclist uses both legs, approximately 200 J of work are done each revolution. Dividing this work by the .83 period results in a power output of approximately 240 W per cycle.
PART 6: ESTIMATING THE MAXIMUM SPEED OF THE CYCLIST Ask the students what additional information they will need to compute the maximum speed that the cyclist can have on level ground. Since they now know the maximum work and power the cyclist can deliver to the bike, this might be a good time to talk about work and its transformation into kinetic energy. Ask the students what factors will limit the speed of the bicycle and its rider. Some students will usually respond that air and road friction ultimately limit the speed. Build upon these reponses by pointing out that when the resistive forces cause a power loss equal to the power input from the cyclist (in this case, 240 W), all of the work being delivered by the cyclist will go into work against the frictional forces and there will be no work which can be transformed into giving the bike and rider additional velocity. Now step through frames 6752-6777 and ask the students if they can determine the speed which Karl would have on this bike if it were on level ground. Review the concept of angular velocity and its relation to linear velocity for a wheel rolling along a level surface.
Another important concept to review is the relationship between force, velocity and power. It may be worth deriving the relationship between these quantities i.e. for constant velocity the power output is the force times the velocity.
Knowing the diameter of the wheel (see frame 52097 which shows it to be 0.34 m) and the number of revolutions it makes per second, it is possible to determine the speed of the bicycle.
First, position the clip to frame 6752. Note at the bottom of the wheel is a small strip which has been placed on the wheel. Have the students mark the center of this strip using an overhead transparency pen. Now have them advance the clip until this strip reappears in the same postion as the mark they made. This will occur at frame 6767 after two revolutions of the wheel. Thus fifteen frames have elapsed or 15/30 seconds for two revolutions of the wheel. Have them calculate the period of rotation of the rear wheel. Most students will easily obtain a period (T) of 0.25 seconds.
Now have the students calculate the angular frequency in radians per second.
Now have the students compute the linear velocity of the bikev = wr = .34 m * 25 rad/s = 8.4 m/s
This speed in meters per second is equal to 16.3 knots, a nautical unit of speed often used to express wind speeds.Students can compute the average force which Karl delivers to the bike during each cycle:
using the fact that P ave = F ave v . students should get Fave = 30 N.
PART 7: STUDENTS COMPARE THEIR CALCULATIONS WITH REAL WIND TUNNEL DATAReal wind tunnel data for Karl on the 12 speed sport bike can be found on the video disc at frame ...The data shows that there was a drag force of about 30 N in a wind speed of 16.5 knots or roughly 30 km/h.
Thus the calculations based on using an average value of the force between crank positions calculated from Professor Hull's dynamotor agrees reasonably well with the experimental wind tunnel data.