The Physics
Hypertextbook
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 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.

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: 0,705 m
• 400 km/h: 2,825 m
• 500 km/h: 4,415 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 the Astronomical Data page at physics.info and The Physics Factbook at its sister web site, hypertextbook.com. 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. trajectories-circular.pdf
The drawings on the accompanying pdf show a mass on the end of a string as it is spun counterclockwise in a vertical circle. A pair of scissors is used to cut the string cleanly and instantly at four different positions. Sketch the subsequent trajectory of the mass until it lands on the ground.
2. Which device(s) on a car can be used to control its speed? Which device(s) on a car can be used to control its velocity but not its speed?
3. A car driving on a circular test track shows a constant speedometer reading of 200 km/h for one lap.
1. Describe the car's speed during this time.
2. Describe its velocity.
3. How do the two compare?
4. In an unusual move by the New York Department of Transportation, all of the "speed limit" signs in the state 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?
5. 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. (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, and
3. at the lowest point.
6. 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.
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.

### 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 (i.e., double, triple, half, one-eighth, the same)?
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, and
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. 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.

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. multiples of the distance from the Earth to the moon
8. Space stations don't need to be round to have artificial gravity. The tethered spacecraft in the diagram to the right 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?
9. 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)
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 1.60 left 0141 012
11 → 12 12 06 0.54 backward 0100 011
12 → 10 06 06 3.60 right 0031 005
totals → 1442 134

### 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.)