The Physics
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Opus in profectus

# Centripetal Force

## Problems

### practice

1. A 250 kg motorcycle is driven around a 12 meter tall vertical circular track at a constant speed of 11 m/s.
1. Determine the normal and friction forces at the four points labeled in the diagram below.
1. at the bottom (and rising)
2. halfway to the top
3. at the top
4. 45° past the top
2. Determine the minimum coefficient of static friction needed to complete the stunt as planned.
(The mass of the motorcycle includes the mass of the rider. Assume that aerodynamic drag and rolling resistance are negligible.) 2. Ringworld (paid link) is the title of a classic science fiction novel written by Larry Niven in 1970. Set in the year 2850, it is the story of four adventurers (two human and two alien) who are chosen to explore an engineered world encircling a sun-like star. The Ringworld is an enormous cylindrical band with a radius roughly equal to that of the Earth's orbit and a width about the same as the diameter of the Sun. It was constructed by some unspecified form of matter transmutation using the planets and minor bodies that once orbited the Ringworld's sun as raw material. The flat, inner surface is covered with a natural-looking, Earth-like terrain and it spins at a speed fast enough to provide its inhabitants with the sensation of Earth-like gravity. Thousand mile high walls along the edges keep the Ringworld's atmosphere from spilling out into space. The Ringworld is the home of hundreds of hominid species, but they are mostly non-technological. The advanced civilization that engineered the Ringworld collapsed centuries ago and the adventurers find only its remains.

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How fast does Ringworld spin to provide its inhabitants with the sensation of normal Earth gravity? State your answer in…
1. meters per second
2. Earth days per rotation
3. rotations per Earth year
3. The following passage outlines the design specifications of a proposed maglev train system (the Transrapid).

The curve radii of modern high-speed systems result in dependence on the speed and the maximum possible superelevation of the guideway to compensate for the centrifugal forces occurring. The Transrapid's guideway can have a maximum superelevation of 12 degree (up to 16 degree in special cases) which allows smaller radii at higher speeds than in the case of conventional wheel-on-rail systems.

• 200 km/h:0705 m
• 400 km/h: 2825 m
• 500 km/h: 4415 m

ThyssenKrupp Transrapid GmbH, 2002

Determine…

1. the maximum centripetal acceleration (in m/s2 and g) implied by these specifications
2. the speed limit (in m/s and km/h) on a curved section of track with the minimal radius
4. Complete the following table.
1. Complete the first two columns using astronomical data from a reliable source. Be sure to specify the units used for each entry.
2. Complete the last two columns using a calculator. Be sure to state your answers in SI units.
Orbital parameters
Moon
Mercury
Earth
Pluto
Sun

### conceptual

1. In an unusual move by the New York State Department of Transportation, all of the "speed limit" signs were replaced with "velocity limit" signs.
1. What would such a sign look like?
2. How could one travel faster than the old speed limit without violating the new velocity limit?
2. Which device(s) on a car can be used to change…
1. its speed?
2. its velocity but not its speed?
3. A car driving on a circular test track shows a constant speedometer reading of 100 kph for one lap.
1. Describe the car's speed during this time.
2. Describe its velocity.
3. How do the speed and velocity compare?
4. Is it possible for an object to have…
1. constant speed and changing velocity
2. changing speed and constant velocity
5. Why are the devices in cars called speedometers and not velocitometers?
6. Draw a free-body diagram for each of the following situations:
1. a car turning a corner on level ground
2. a model airplane on the end of a string, flying in a horizontal circle
3. a roller coaster at the top of a vertical loop where the roller coaster is upside-down
4. a car rounding a banked curve
5. a pendulum released from a 60° angle at three points in its motion…
1. immediately after it's been released
2. halfway to the bottom
3. at the lowest point
7. The Physics Teacher has published several articles containing free body diagram worksheets. They are available free to members of the American Association of Physics Teachers (AAPT). Everyone else has to pay.
8. When cars turn a corner, they often follow a path that is an arc of a circle. Identify 3 quantities that could vary between cars and drivers as they turn a corner. How might changing each of these quantites separately affect the magnitude of the centripetal force needed to turn the car? Do not identify factors that have no effect.
9. One fine day as I was sitting in the passenger seat in my friend'sPOS car, he made a left turn and the door popped open. I was rewarded with a closeup view of the moving pavement as my upper body spanned the ever widening gap between the seat and the distant arm rest. I calmly pulled on the door handle, straigtened myself back up into a seated position, and shut the door. On the diagram below, sketch the path I would have taken had I not been wearing a seatbelt. ### numerical

1. A 500 kg race car rounds a curve with a radius of 100 m.
1. What type of force provides the centripetal force in this example?
2. Find the magnitude of the centripetal force acting on the car when it rounds the curve at 20 m/s.
3. Find the magnitude of the centripetal force acting on the car when it rounds the curve at 60 m/s.
4. How does the centripetal force at 60 m/s compare to the centripetal force at 20 m/s. (Is it double, triple, half, one-eighth, the same, something else?)
2. Some people rejected the notion that the Earth is rotating when it was first proposed. Since the Earth is so large, points on the equator would be moving quite fast and it was thought that objects on the equator would be flung off into space. Show that the acceleration due to gravity is more than sufficient to keep this from happening through the following calculations.
1. Find the speed of a point on the equator.
2. How does this speed compare to the speed of sound in air?
3. Find the centripetal acceleration needed to remain on the equator.
4. How does this centripetal acceleration compare to the acceleration provided by gravity?
3. A cylindrical space station of diameter 500 m is set spinning to provide the sensation of normal Earth gravity. Determine…
1. the speed of a point on the floor of the space station
2. the time to complete one revolution
3. the number of revolutions per minute
4. In 1959, R. Flanagan Gray, a physician at the Aviation Medical Acceleration Laboratory in Johnsville Pennsylvania, subjected himself to 31.25 g of transverse acceleration for five seconds. This performance, in a water-filled aluminum capsule incorrectly nicknamed the "Iron Maiden", established a new record for centrifugal acceleration tolerance. Given that the capsule was positioned 15 m (50 feet) from the center of rotation, determine…
1. the speed of the capsule
2. the period of rotation
3. the number of rotations during the five seconds of peak acceleration
5. A stunt motorcycle track has a section which is a vertical loop of radius 5.0 m. At what minimum speed should a motorcycle be driven through…
1. the top of the loop?
2. the bottom of the loop?
6. A 0.10 kg solid rubber ball is attached to the end of an 0.80 m length of light thread. The ball is swung in a vertical circle. The speed of the ball is kept constant at 6.0 m/s throughout this experiment. Determine the tension in the thread at…
1. the top of the circle and
2. the bottom of the circle.
7. It takes a plane flying at 150 km/h 3.0 minutes to circle a cloud at an altitude of 3000 m. What is the diameter of the cloud?
8. Two related questions.
1. Calculate the orbital speed of the moon.
2. How long does it take the moon to move a distance equal to its diameter?
9. Two related questions.
1. Calculate the orbital speed of the Earth.
2. How long does it take the Earth to move a distance equal to its diameter?
10. Geosynchronous, Earth-orbiting space station
For a sufficiently advanced human civilization, the occasional trip into outer space may become a reality for the general population. Having large numbers of spacecraft landing and taking off from the surface of the Earth would probably not be acceptable, however. One way around this would be to dismantle the moon and use it to build a ring around the Earth that rotates at the same rate as the Earth. This ring would be linked to the equator by electrically powered space elevators. No more noisy, dirty rockets. Just hop on the space elevator, press the "up" button, and stare nonchalantly at the door for a couple of days. Such a massive structure would also house a large population of full-time inhabitants. Since they're the descendants of Earth-bound humans, they would probably feel most comfortable in a 1 g environment.

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Determine the radius of such a megastructure. (Assume the acceleration due to gravity is negligible at this distance from the Earth.) State your answer in terms of…
2. fractions of the distance from the Earth to the moon
11. Space stations don't need to be round to have artificial gravity. The tethered spacecraft in the diagram below represents an alternative design. Put the astronauts in a passenger capsule. (Put the "spam in a can" as they say.) Tether it to a heavy counterweight 16 km away and spin the whole thing around once every three minutes. Place your docking port and nuclear power plant on the axis and you're all set. 1. How far away should the passenger capsule be placed from the axis of rotation so that it experiences an apparent gravity equal to normal Earth gravity?
2. What is the apparent gravity at the counterweight?
3. What is the apparent gravity at the nuclear power plant?
12. The diagram below shows the plan of a go-kart track. All the straightaways are straight and all the curves are arcs of a circle. None of the turns are banked. The twelve numbered positions are junctions between straightaways and curves (or curves and straightaways). The spacing between grid lines is 10 m. Any height changes are minor can be ignored when performing calculations. The table below shows the instantaneous speed of one particular go-kart at the beginning and end of each segment. Determine the following quantities to the stated precision and complete the table.

1. magnitude of acceleration (to the hundredths place)
2. direction of acceleration (forward, backward, left, right) relative to the direction of motion
3. distance (to the nearest meter)
4. duration (to the nearest second)
segment

initial speed
(m/s)
final speed
(m/s)
acceleration
(m/s2)
acceleration
(direction)
distance
(m)
duration
(s)
1 → 2 06 08
2 → 3 08 08
3 → 4 08 12
4 → 5 12 12
5 → 6 12 16
6 → 7 16 16
7 → 8 16 06
8 → 9 06 06
09 → 10 06 12
10 → 11 12 12
11 → 12 12 06 0.54 backward 0100 011
12 → 10 06 06 3.60 right 0031 005
totals → 1442 134
13. A 5 g coin is placed on a horizontal turntable 15 cm from its center as shown in the diagrams below. The turntable rotates at a constant rate in a counterclockwise direction as seen from above. The coin does not slip and the turntable completes one rotation every 800 ms.
1. On the figure labeled "side view", draw and label arrows coming out of the coin to represent the forces acting on the coin. (You do not need to draw the arrows to scale.)
2. On the figure labeled "overhead view", draw and label arrows coming out of the coin to represent the direction of the instantaneous acceleration and velocity of the coin. (You do not need to draw the arrows to scale.)
3. Determine the magnitude of the instantaneous velocity of the coin.
4. Determine the magnitude of the instantaneous acceleration of the coin.
5. Determine the weight of the coin.
6. Determine the normal force between the coin and the disk.
7. Determine the friction force between the coin and the disk.
8. If the rate of rotation of the turntable is increased by even the slightest amount, the coin slips off the turntable. Determine the coefficient of static friction between the coin and the platform.
14. The polyester thread used in a physics lab is rated to withstand 20.0 pounds of tension (89.0 newtons) before breaking. An out of control student decides to spin a 0.034 kg rubber stopper at the end of 0.851 m of fishing line in an attempt to break it. The student whips the stopper around in a horizontal circle so that it makes one complete revolution in 0.125 s.
1. What is the magnitude of the stopper's velocity?
2. What is the tension in the thread?
3. Does the student succeed in breaking the thread?
15. Read the following excerpt from the US print and web publication, Popular Mechanics, before beginning this problem.

The Bugatti Chiron is high-performance excess at its best. It's a 1479 hp beast that's quicker than many race machines, and is limited to 261 mph. But more than impressive big numbers is the smart engineering that lets this supercar manage all that power and speed. After spending a few hours with Bugatti test driver and Le Mans champ Andy Wallace in the near \$3 million machine, here's what I learned about the car and the true meaning of "supercar"….

Bugatti says it won't attempt a true top speed run until 2018. Wallace implied the Chiron reaches its limited speed of 261 mph with relative ease. So, could there be another 39 mph left? He says that speed is unlikely because the forces at that level are devastating.

Even at 261 mph the wheel and tire has to withstand extreme forces. The valve cap on each wheel weighs 2.5 grams, but it equates to 16 pounds at 261 mph. As the speed moves even higher, the loads increase exponentially. Wallace says at the moment, there is no tire that can withstand the g-loading at 300 mph, but he predicts Michelin engineers will sign off on a top speed with the current production tire slightly north of 280 mph. Perhaps a future version of this car will be engineered to hit that magic 300 mph milestone.

Ben Stewart, 2017

Bugatti Chiron tire specifications* * David Tracy at jalopnik.com reported these tire sizes, but I had trouble finding data for the Michelin brand. Data for similar Pirelli's from Amazon.com (paid links) are used in their place.
tire size width inner
diameter
outer
diameter
mass max. inflation
pressure
front 285/30R20 285 mm 20 in. 26.8 in. 27.0 lbs 50 psi
rear 355/25R21 355 mm 21 in. 28.1 in. 31.7 lbs 50 psi

Please note: this problem is full of mixed units — something that is still common among US automotive engineers in the 21st century. Use an online calculator that can handle mixed unit computation like this one does. Or convert all the relevant quantities to SI units before doing any calculations.

1. Determine the angular speed of the rear tire at 261 mph.
2. Determine the translational speed of a point on the inside rim of the rear tire at 261 mph.
3. Determine the centripetal acceleration of a point on the inside rim of the rear tire in…
1. m/s2 (the International System unit for acceleration)
2. g's (multiples of the acceleration due to gravity on Earth)
4. Use your results from c. ii. to determine the apparent weight of the valve cap. Do your results agree with those claimed in the article? If not, why not?
5. Since you live in the article's future, respond the question implied by its final sentence.
16. The Canterbury Velodrome in Sydney, Australia is an oval racing track for bicycles with 71.220 m straightaways and 95.445 m turns. An exceptional cyclist is able to complete one lap in 18.62 s. Determine the…

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1. length of the track
2. average speed of the exceptional cyclist over one lap
4. centripetal acceleration of the exceptional cyclist in a turn
17. The Rotor is an amusement park ride that consists of a cylindrical, padded room with a radius of 2.46 m that spins once every 1.8 s. A 90 kg person steps inside. Once the Rotor gets up to speed the floor moves down and the person remains "stuck" to the wall.
1. On the figure labeled "side view", draw and label arrows coming out of the person to represent the forces acting on the person. (You do not need to draw the arrows to scale.)
2. On the figure labeled "overhead view", draw and label arrows coming out of the person to represent the direction of the instantaneous acceleration and velocity of the person. (You do not need to draw the arrows to scale.)
3. Determine the magnitude of the instantaneous velocity of the person.
4. Determine the magnitude of the instantaneous acceleration of the person.
5. Determine the weight of the person.
6. Determine the normal force between the person and the Rotor.
7. Determine the friction force between the person and the Rotor.
8. Determine the minimum coefficient of static friction between the person and the Rotor needed for the person to remain "stuck" to the wall.
18. Richmond Raceway is a motorsport racing, short track located outside Richmond, Virginia. Not a true oval, the track could be described as D-shaped or tri-oval (a cross between a triangle and an oval). Using the data table and diagram below, calculate…
1. the total length of the track
2. the centripetal acceleration in each turn at the average speed for that turn
3. the time to complete each segment of one lap given the average speed for that segment
4. the time to complete one lap
5. the average speed for one lap
A car driving one lap on the Richmond Raceway
turn
1 & 4
turn
2 & 3
back
stretch
total
length
(m)
196.60 275.84 262.13
(m)
365.36 108.51 n/a n/a
banking
(°)
8 14 2 n/a
average
v (m/s)
101.03 66.99 82.12
average
ac (m/s2)
n/a n/a
time
(s)

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### algebraic

1. A rock of mass m is tied to a string and spun in a vertical circle of radius r at a constant speed. At the top of the circle, the tension in the string is twice the weight of the rock. Determine the following quantities in terms of g, r, and m
1. the tension in the string at the top of the circle
2. the speed of the rock at the top of the circle
3. the speed of the rock at the bottom of the circle
4. the tension in the string at the bottom of the circle
2. As a highway engineer, you wish to design a safe curve for a highway with a speed limit v of 24 m/s (54 mph). Rubber tire on dry pavement has a coefficient of static friction, μs, of 0.75.
1. What is the relation between the radius r of a turn and the known quantities in this problem for a car that is not skidding out of control? That is, state r as a function of v, μs, and g. (Note: a variety of vehicles with different masses will be traveling on this highway. Somehow you must eliminate mass from your equation.)
2. Given the numbers in this problem, determine the radius of a curve that is just safe enough to allow a car traveling at the speed limit to safely round the corner.
3. Engineers often "over design" their projects to reduce the probability of failure. For example, bridges are built many times stronger than is necessary to just support the weight of traffic. Name at least two things that should be done to ensure that this highway curve is over designed.
3. A problem for Americans and other people who like old-fashioned units. Follow this link for a hint (assuming this webpage has not vanished).
1. What is the equation on this piece of paper all about? (This is the easier question.)
2. What is the significance of the number 15? (This is the harder question.) 