3 Rides
In this section we will discuss how you can apply physics principles to some of the rides at the South Carolina State Fair. Analyze several of the rides mentioned in this booklet and then use those rides as starting places for investigating other rides.
One of the measurable aspects of fair rides is the force that they exert on you and the acceleration you feel. The force and acceleration are connected through Newton's second law; F = ma, where F is the force in Newtons (N), m is the mass in kilograms (kg), and a is the acceleration in meters per second per second (m/s
2). When the only force on you is the force of gravity your weight is W = mg, where g is the acceleration of gravity (9.81 m/s2). When you feel "pushed down" at the bottom of a roller - coaster loop you experience an apparent weight greater than your normal weight. When your apparent weight is twice your normal weight we say you experience a force of "two g's". That is because you experience the acceleration of gravity plus the acceleration of the ride's motion. At another point on the ride you could be almost lifted from your seat. That is an example of less than one g because you feel lighter than your normal weight. The downward acceleration of gravity is the same, but your acceleration due to the ride cancels some of the effect of gravity. One of the objects of ride physics is to measure and calculate the acceleration of a rider.
3.1
Roller Coasters; Non-looping and Looping, ClothoidsA roller coaster provides a good example of conservation of energy. The train is pulled to the top of a high hill and let go (Fig. B1). The train then plunges down the other side of the hill. It then rises up another hill to plunge down again, perhaps with twists and turns to the left and right.
We can understand the motion of a roller coaster by considering conservation of mechanical energy. Assume the car is hauled up to point A. At point A the total mechanical energy is equal to the total mechanical energy at point E, as it is at all of the points on the track. (For the purpose of getting a general understanding, we are omitting the energy converted into heat by friction. You may want to try to estimate the magnitude of frictional effects in a more refined analysis. However you can best understand things by leaving frictional effects out at the start.) Thus the sum of the potential energy (PE) and kinetic energy (KE) is the same at points A and E.
Therefore we can calculate the speed at E if we know the heights h
A and hE. If the train is moving slowly at the top of the first hill (A) we can neglect the term
To make the ride more exciting the roller coaster designers may put a loop - or even two loops- in the track, as shown in Fig. B2. They must choose the maximum height of the loop, and also
the shape of the loop. You can estimate the maximum height of the loop, or to turn the question around, can you estimate what fraction of the initial potential energy at A is lost to friction if the car has a given speed at position E and height h
E.If the loops were circular the riders would experience 6 g's at the bottom and would just barely hang on (0 g's) at the top of the loop. This is under the assumption that the coaster enters the loop at a given speed at C and is slowed by the force of gravity as it coasts up to the top of the loop at D. Humans can not tolerate 6 g's in this situation so another shape rather than a circle is used for the loop.
The shape that is used is part of a spiral called a clothoid. A clothoid spiral (Fig. B3a) is useful for joining trajectories and is sometimes used in highway and railroad interchanges.
3.2 Rotating Rides and Swings
Merry-Go-Round
Perhaps the simplest rotating ride is a merry-go-round. You can still make some interesting measurements on it. Figure B4 is a schematic diagram of a merry-go-round with two rows of horses. Here we will not include the up-and-down motion of the horses, so make your measurements standing on the rotating platform.
The horizontal component of the centripetal acceleration of is a c = v2/r for a horse at a distance r from the axis of rotation and moving at a speed v.
In terms of the period T - the time for one complete rotation - we can write
Because the period T is the same for all horses, no matter what their distance from the axis, we can write the ratio of the centripetal acceleration at two different distances r
1 and r2 as
You should determine r
1 and r2 and measure ac(1) and ac(2) with your horizontal accelerometer and see how your measurements compare with the formula above.
Rotator
Another simple ride is sometimes called Rotator .(Fig.B 5) Riders stand with their backs against the wall in a cylindrical chamber that can be rotated about its vertical centrakl axis. When the ride has
reached a certain speed the floor drops down and the riders are held against the wall by frictional forces. Can you estimate the minimum coefficient of friction between the riders and the wall needed to keep them from falling?
Round Up
Round Up or they may be in cars or cages, as in the Enterprise. Taking the Round Up as an example, (Fig. B.6) note that in the horizontal position the acceleration -or force- is just that of the Rotator. As the circling ride is tipped up the force changes as you go around because the force exerted on you due to the ride is always toward the center of rotation but the force of gravity is always downward. We can write
,where F is the net force you feel while riding the Round Up, Fwall is the constant towards -the-center force on you by the Round Up supports, and Fgravity is the force of gravity (inward at the top and outward at the bottom).
Thus for a completely vertical Round-Up with constant angular speed the acceleration away from the center at the top would be
,and the acceleration away from the center at the bottom would be
.
To be more precise we note that the Round-Up is tilted up to an angle
where the minus sign is for the top and the plus sign is for the bottom.
Swings
What does this formula predict about the angles of the riders compared to the angles of the riderless swings? Is the answer the same for the Wave Swinger, which has a tilted axis of rotation?
3.3 Compound Motion Rides
about an axis which itself moves in a circle. On some rides there also may be up-and-down motion or in-and-out motion. Observe the rides and make a rough sketch. Determine where the maximum change in velocity will occur and check your predictions by making measurements with your accelerometer.
3.4 Drops
There are at least two types of "drops." One is similar to a parachute ride and the other is close to a free-fall. For either case you should be able to think of some measurements and observations. The parachute type has been around for some time and the free-fall type is now being brought to fairs. The rider is droped from the top of a tall tower between rails that are nearly friction free. You should measure the height of the fall and then calculate the speed at the bottom. You can also calculate the expected time for a friction-free fall. Then measure the final speed or time-of-fall and comment on how friction-free the ride is.
3.5 Ride Data
Here we have listed some, but not all, of the parameters of several rides. You will need to make some measurements for yourself.
G - FORCE
DOPPLE LOOP
Double looping roller coaster.
Height of first drop: 104 ft.
Height of loop: 45 ft.
ENTERPRISE
Tilting circular ride.
Diameter of wheel: 55 ft.
Rotation speed: 13.5 rpm.
Angle with the horizontal: maximum 87°.
POLAR EXPRESS OR THE HIMALAYA
Tilted compound motion.
Radius of the ride: 20 feet
Rotation speed: 4.8 seconds per revolution
Height gain to highest point of ride: 9 feet
GIANT WHEEL
A Ferris wheel
Radius: 52 ft.
ROUND UP
RAINBOW
A rotating horizontal platform.
Speed: 9-9.5 revolutions per minute
Length of arm: 30 ft.
WAVE SWINGER
Rotating swings.
Length of chain: 16 feet, 6 inches
Radius of rotation: 37 feet, 2 inches
Speed: 6 seconds for one revolution
WILD CAT
Corkscrew roller coaster.
Length of track: 560 m
Time for ride: 95 seconds
Length of car: 7 feet, 9 inches
Weight of car: 800 lb, holds 4 people
Length of first drop: 77 feet
Time for the first drop: 2 seconds
Height of the first hill: 46 feet
Radius of the first curve: 27 feet, 2 inches