GETTING STARTED
MAKING MEASUREMENTS
Time:
The times that you are required to work out problem can be measured using a digital watch with a stopwatch or with a second hand. When measuring the period of a ride that involves harmonic or circular motion, measure the time for several repetitions of the motion. This will give a better estimate of the period of the motion that just measuring one repetition. In any case, measure multiple occurrences and then the average.
Distance:
Because of the locations and normal operation of the rides, you will not be able to directly measure heights, diameters, etc. All but a few of the distances can be measured remotely using one or another of the following methods. They will give you a reasonable estimate. Consistently use one unit of distance – meters of feet.
1. Pacing: Determine the length of your stride by walking at your normal rate over a measured distance. Divide the
distance by the number of steps – thus giving you the average distance per step. Knowing this, you can pace off horizontal distances to help with calculating the heights of rides.
2. Ride Structure: Distance estimates can be made by noting regularities in the structure of a ride. For example, tracks may have regularly spaced cross members. The length of the track can be estimated by estimating the length of one of the regularly spaced cross members of the track and then multiplying this by the number of cross members of track. This can be used for both vertical and horizontal distances of track.
3. Triangulation: A horizontal accelerometer can be used as a sextant to measure the height (h) of a ride. Triangulation
where you cannot measure the distance to the base of the ride may be done by sighting the top of the ride from two
different positions on the ground. Sight the top of the ride from one location and obtain one angle. Pace off a distance D
toward the ride and sight the same part on the ride to obtain another angle. Use the Law of Sines to obtain the height of
the ride.
Speed:
The average speed of an object is simply the distance the object travels divided by the time it takes to travel that distance. To calculate the average speed of an object traveling in a circle , divide the circumference by the time for one revolution. To measure the speed of a train as it passes a particular point, time how long it takes the train to pass the chosen point. You may have to estimate the length of one car on the train and then multiply this length by the number of cars in the train to get an estimate of the train’s length.
ON YOUR WAY TO THE PARK
PART A: STARTING UP
Things to Measure:
As you pull away from school or from a stop light, find the time it takes to go from stopped to 20 miles per
hour. You may have get someone up front to help out on this. T = ______seconds.
Things to Calculate:
Show equations used and unit cancellations:
1. Convert 20 mph to m/s. (1.0 mph = 0.44 m/s).
v = ______m/s.
2. Find the acceleration of the bus in m/s².
a = ______m/s
3. At a second stop, set up your accelerometer and record the maximum acceleration that the bus experiences.
a = ______m/s
4. Using your mass in kilograms, calculate the average force on you as the bus starts up. (1 kg of mass
weighs 2.2lbs.).
F = ______N.
5. Using your mass in kilograms, calculate your weight in Newtons. (W = mg).
W = ______N.
6. How does this force (#3) compare to the force gravity exerts on you (your weight in Newtons – (#4))?
Circle one: (more, less )
7. How many g’s are you experiencing during the bus’ acceleration?
# g’s = Force experienced (#3) .
Force gravity normally exerts (your weight in Newtons)
#g = ______g’s.
PART B: GOING AT A CONSTANT SPEED
Things to Notice:
8. Describe the sensation of going at a constant speed. Do you feel as if you are moving? Why or whynot? (You will have to try to ignore the effects of the bumps in the road).
9. Are there any forces acting on you in the direction that you are moving ? Explain what is happening interms of Newton’s First Law.
PART C: ROUNDING CURVES
Things to Notice:
10. If your eyes are closed, how can you tell when the bus is going around a curve? Try it and report whatyou notice.
11. As the bus rounds a curve, concentrate on a tree or a building that would have been straight ahead.
See if you can sense that you are going straight but are being pulled into the curve by a centripetalforce.
12. What is supplying the centripetal force that is helping you round the curve?
13. How does this change when the curve is tighter or if the bus is going faster?
RIDES THAT GO IN HORIZONTAL CIRCLES
DATA:
Time for 10 revolutions of the Carousel®: ______seconds.
Angle measured by horizontal accelerometer: ______degrees.
Acceleration measured using accelerometer: ______m/s².
QUESTIONS AND CALCULATIONS:
1. Where would you need to sit if you wanted to experience the greatest speed? Why?
2. If you were on a horse and were to drop a coin while the ride was in motion, where would it land relative to you? Describethe path of the coin until it reaches the floor.
3. Now suppose you stood near the edge of the Carousel® and dropped the coin off the side. When the coin hits the ground,where would it be relative to you?
4. Would you ride the Carousel® on the outermost horse if the frequency of the Carousel® were 1 Hz? Why / Why not?
5. Which direction would you imagine the force acting on you would be trying to push you…toward the center of the Carousel®or away from the center of the carousel? Is there really force acting on you in this manner? What is the cause of feeling thisway?
6. Find the frequency in rpm’s (revolutions per minute) and in hertz.
7. Calculate the linear (tangential) velocity of your horse.
8. Calculate your centripetal acceleration. Would it be the same for all horses or does it changes from center to outer horse?
9. What is the direction of the net force acting on your horse?
10. Notice the way the floor tilts. What could be the reason for this tilt?
11. Your mass in kilograms is ______(found from the ON YOUR WAY TO THE PARK section of this packet) and your weightin Newtons is therefore ______. What is the net force acting on you?
12. What is your tangential acceleration?
13. What is your kinetic energy?
RIDES THAT GO IN HORIZONTAL CIRCLES
BEFORE YOU RIDE:
Locate the large horizontal curve. As you stand facing the ride (at the entrance to the ride’s queue line), the large horizontalcurve will be to your right. You may need to observe the train as it travels to locate this curve.
DATA:
Your mass = ______kg.
Your weight = ______N.
Time for entire train to pass point P at the side of the loop = ______s.
Angle of the train to the vertical () = ______degrees (estimated)
Length of train = 12m.
What would happen to the angle of the train, , if the train were moving faster?
How would be affected if the radius of the curve were larger?
R = 8.5m
WHILE YOU RIDE:
Observations: While going around the curve…
1. Sensation: Circle One: (Normal, Heavier, Lighter)
2. On what part of your body did you feel forces being exerted as you rounded the curve? ____
3. Even though the train was at such a great angle as it came around the curve, did you ever feel as if you were falling out?
4. Explain.
Maximum Acceleration Value = ______m/s2.
QUESTIONS AND CALCULATIONS:
(while rounding the large horizontal curve)
5. Use the length of the coaster and the time it took to pass point P (in the previous diagram) to calculate the average speed ofthe coaster as it rounds the far turn. v =______m/s.
6. What type of force keeps you going around the curve?
7. What provides you with this force?
Two forces act on you as you ride, your weight and the seat force. They are shown atthe right in bold. The seat force has two components. The vertical component balancesyour weight. The horizontal component provides the centripetal force needed tomake you follow the arc of the turn. Combined as vectors they give the force you werefeeling.
8. Calculate the centripetal force on you. Fc = mv² / r.
Fc = ______Newtons.
9. Draw to scale on the diagram given.
a. your weight (in Newtons) pointing down.
b. the vertical component of the seat force pointing up (it is the same size asyour weight!)
c. the horizontal component which is the centripetal force you calculatedabove.
10. Complete the vector diagram: Find the resultant
a. by approximating the length and angle of the on the diagram and
b. by calculation:
Seat Force = ______N @ angle ()______degrees.
11. How does your calculated seat force angle compare to the angle and seat forceyou measured? How can you account for discrepancies?
RIDES THAT GO IN VERTICAL CIRCLES
BEFORE YOU RIDE:
Estimate the diameter of the Wheelie® by triangulation or by scaling with a known sized object:
r1 = ______degrees 2 = ______degrees h = ______meters
Diameter = ______meters
Radius = ______meters
1. Note the Angle each car makes with the vertical as the wheel approaches full speed while still rotating horizontally. Is each caruniformly the same angle, regardless of its position around the wheel?
Carefully observe the angle of each car relative to the suspension point as it rotates after the Wheelie® is completely raisedvertically.
2. Why is it different when approaching the very top from when it is approaching the very bottom?
DATA:
Time for 10 revolutions (when traveling at full speed): ______seconds
Period (at full speed): ______seconds
Frequency (at full speed): ______hz
WHILE YOU RIDE:
Accelerometer readings: (Best estimate from time stamp)
a. at full speed, but while still horizontally oriented: ______g’s.
b. at the top: ______g’s.
c. halfway down: ______g’s.
d. at the bottom: ______g’s.
e. halfway up: ______g’s.
QUESTIONS AND CALCULATIONS:
3. Calculate the circumference of the ride. Then calculate the linear (tangential) speed of the car at full speed, but still horizontallyoriented.
v = ______m/s.
4. What is your centripetal acceleration at this phase of the ride?
ac = ______m/s2
5. How does your calculated result compare to the measured value?
6. Draw a free body diagram showing the forces acting on you:
a. at the top
b. at the bottom
c. halfway down
7. Where is the net force the greatest? Is this calculation collaborated with your accelerometer data? How did you feel at thispart of the ride?
RIDES THAT GO IN VERTICAL CIRCLES
BEFORE YOU RIDE:
*Use triangulation to determine the height (h1) of the first hill and the height (h2) of the largest loop.
r1= ______degrees (hill) r2 = ______degrees (hill)
r1= ______degrees (loop) r2 = ______degrees (loop)
h1 = ______meters h2 = ______meters
Step back and look carefully at the loop. It is not really circular. For safety’s sake, the car is going faster than the absolute
minimum speed at the top of the loop – so even when friction is a factor, this additional speed would make the rider uncomfortableat the bottom of the loop is avoided by using what is called a clothoid loop in place of a circular loop.
1. Sketch the loop asyou see it.
2. The radius of curvature is changing. Does this increase or decrease as you go from the bottom to the top of the loop?
WHILE YOU RIDE:
Use your accelerometer to measure the g’s at the various points during the ride.
Minimum acceleration ______g’s experienced at the (bottom, midway, top) of the loop.
Maximum acceleration ______g’s experienced at the (bottom, midway, top) of the loop.
QUESTIONS AND CALCULATIONS:
3. What happens to the gravitational potential energy on your way down the first hill?
4. Describe (in detail) the transition between gravitational potential energy and kinetic energy as the coaster goes through aloop.
4. What two forces are acting on the train at the top of the loop? Give the direction of each force.
5. Why doesn’t the rider fall out of the train at the top of the loop and land on his head?
6. Using the data collected above, calculate the value of the gravitational potential energy at the top of the highest hill and atthe top of the loop.
7. From conservation of energy considerations use the change in height from the top of the first hill to the top of the loop tocalculate the speed of the train as it goes through the highest point in the loop. Does this satisfy the critical speed criteria? Vc =√(rg)
8. Assume that the loop is circular. From your measure of the height – calculate the critical velocity and the kinetic energy atthe top of the loop.
9. From the potential energy calculated in #1 and from the kinetic energy calculated in #2, calculate the velocity of the train atthe top of the loop. Does the train have enough speed to travel in a circle?
10. What force would be exerted on you at the bottom of the loop if the loop were circular? How many g’s would this be?
Since most people begin to feel uncomfortable beyond 3.5 g’s and some even pass out just over 4.0 g’s, do you see why aclothoid rather than a circular loop is used?
RIDES THAT GO IN VERTICAL CIRCLES
BEFORE YOU RIDE:
*Use triangulation to determine the height (h1) of the first hill and the height (h2) of the largest loop.
r1= ______degrees (hill) r2 = ______degrees (hill)
r1= ______degrees (loop) r2 = ______degrees (loop)
h1 = ______meters h2 = ______meters
Step back and look carefully at the loop. It is not really circular. For safety’s sake, the car is going faster than the absolute
minimum speed at the top of the loop – so even when friction is a factor, this additional speed would make the rider uncomfortableat the bottom of the loop is avoided by using what is called a clothoid loop in place of a circular loop.
1. Sketch the loop asyou see it.
2. The radius of curvature is changing. Does this increase or decrease as you go from the bottom to the top of the loop?
WHILE YOU RIDE:
Use your vertical to measure the g’s at the various points during the ride.
Minimum acceleration ______g’s experienced at the (bottom, midway, top) of the loop.
Maximum acceleration ______g’s experienced at the (bottom, midway, top) of the loop.
QUESTIONS AND CALCULATIONS:
3. What happens to the gravitational potential energy on your way down the first hill?
4. Describe (in detail) the transition between gravitational potential energy and kinetic energy as the coaster goes through aloop.
5. What two forces are acting on the train at the top of the loop? Give the direction of each force.
6. Why doesn’t the rider fall out of the train at the top of the loop and land on his head?
7. Using the data collected above, calculate the value of the gravitational potential energy at the top of the highest hill and atthe top of the loop.
8. From conservation of energy considerations use the change in height from the top of the first hill to the top of the loop tocalculate the speed of the train as it goes through the highest point in the loop. Does this satisfy the critical speed criteria? Vc =√(rg)
9. Assume that the loop is circular. From your measure of the height – calculate the critical velocity and the kinetic energy atthe top of the loop.
10. From the potential energy calculated in #1 and from the kinetic energy calculated in #2, calculate the velocity of the train atthe top of the loop. Does the train have enough speed to travel in a circle?
11. What force would be exerted on you at the bottom of the loop if the loop were circular? How many g’s would this be?
Since most people begin to feel uncomfortable beyond 3.5 g’s and some even pass out just over 4.0 g’s, do you see why aclothoid rather than a circular loop is used?
RIDES THAT ALLOW YOU TO FALL
BEFORE YOU RIDE:
A. Use the triangulation method explained in the “Making Measurements” section of this packet to determine the
height of the tower.
B. Record the time it takes for the ride to fall until the unit reaches the breaking point. You should time the ride for
several falls and use an average time in your calculations.
DATA:
r1 = ______degrees
r2 = ______degrees
Distance paced off (D) = ______meters
h =______meters
Time of fall = ______seconds
Maximum Accelerometer Reading ______
QUESTIONS AND CALCULATIONS
Try to ride Acrophobia® twice (if you can stomach it). The first you ride it look out (away from the ride) and the second timeyou ride it, look down.
1. Which is harder to do? Why?
2. Using the time of fall and the acceleration due to gravity, how fast is Acrophobia® traveling at the moment that the breaksbegin to slow the ride?
v = gΔt = ______m/s
3. The mass of the passenger-carrying ring is 10,542kg. How much kinetic energy does the ring have at the velocity that youcalculated in question 2?
4. Use conservation of energy to determine the height of the tower.
H = ½v2/g.
5. How does this value compare to your measured value in the DATA section above? Account for any discrepancies.
6. Are you moving toward the ground at this rate, or is the ground moving toward you at this rate? Discuss in terms of frame ofreference.
RIDES THAT ALLOW YOU TO FALL