Centripetal Force

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centripetal

discussion

Centripetal acceleration from calculus

[diagram of standard position circle]

r   x =  +r cos (ωt) y =  +r sin (ωt)
 
           
v =  dr   vx =  −rω sin (ωt)     vy =  +rω cos (ωt)  
dt
           
a =  dv   ax =  −rω2 cos (ωt)     ay =  −rω2 sin (ωt)  
dt

Check out the negative sign in acceleration. This shows that acceleration is toward the center at all times (that is, opposite the radius). Harder to see is the 90* phase difference between velocity and acceleration

Note

 r2   =   x2  +   y2
 r2   =   [+r cos (ωt)]2   +   [+r sin (ωt)]2 
 r   =   r 

Note

 v2   =  vx2  +  vy2
 v2   =   [−rω sin (ωt)]2   +   [+rω cos (ωt)]2 
 v   =   rω 

Note

 a2   =  ax2  +  ay2
a2   =   [−rω2 cos (ωt)]2  +   [−rω2 sin (ωt)]2
 a   =   rω2

centrifugal

discussion

rotating reference frame


[magnify]

Summary

Problems

practice

  1. Write something.
    • Answer it.
  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 sufficiently 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

    Solutions …

    1. Use the centripetal acceleration formula and solve for speed. Substitute values for the acceleration due to gravity on earth and the radius of the earth's orbit (also known as an astronomical unit).
       
       ac  =   v2      
       r       
       v   =  √acr   =  √ [(9.81 m/s2)(1.50 × 1011 m)]   
       
       v   =   1.21 × 106 m/s  
       
       
      Sounds fast, but the speed of things in space tends to be fast anyway. How does this compare to the yearly motion of the earth around the sun? That's the goal of the next parts of the question.
    2. We'll solve this practice problem two ways. First we'll use the definition of speed and substitute the value calculated above and the distance traveled in one rotation (the circumference).
       
      v  =  s  =  2πr                    
      t T    
      T  =  2πr  =  2π(1.50 × 1011 m)  = 780,000 s ≈ 9 days
      v 1.21 × 106 m/s
       
      The fancier way is to start from the centripetal acceleration formula, replace speed with circumference over time, and simplify. This gives us a formula that some people like so much they memorize it.
       
      ac  =  v2  =  (2πr / T)2  =  4π2r
      r r T2
       
      Solve for period and substitute.
       
      T  = 2π √  r  = 2π √  1.50 × 1011 m  = 780,000 s ≈ 9 days
      ac 9.81 m/s2
       
      That's a quick "year" (if we use the astronomical definition of the year as the period of one trip around the sun). How does it compare to a regular calendar year?
    3. This problem is best solved by dimensional analysis.
             
      n  =  365 days / year  = 40.5 rotations per year
      9 days / rotation
             
      We will return to this problem in the section on power.
  3. The following passage outlines the design specifications of a proposed maglev train system (the Transrapid).
    Frage/Question: What is the Transrapid's curve radius?
    Antwort/Answer: 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.
    • Minimal radius: 350 m
    • 200 km/h: 705 m
    • 400 km/h: 2,825 m
    • 500 km/h: 4,415 m
    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

    Solutions :

    Once you get past reading the awkward translation from German to English, this is a conceptually easy question. Set up a table like the one below and complete the missing parts. Use a graphing calculator as it eliminates the drudgery of repeated calculation.

    maximum speed radius centripetal acceleration
    (km/h) (m/s) (m) (m/s2) (g)
    – – 350 – –
    200 – 705 – –
    400 – 2,825 – –
    500 – 4,415 – –
    mean centripetal acceleration → – –
    1. Start by converting the maximum speeds from km/h to m/s.
       
      ⎡
      ⎣      		 m  =  km   1000 m   1 h ⎤
      ⎦
      s h 1 km 3600 s
       
      Then apply the formula for centripetal accelration.
         
      ac =  v2
      r
         
      The resulting values will be in m/s2. To convert to g, divide by the standard value for the acceleration due to gravity.
       
      ac[g] = ac[m/s2] ÷ 9.80665 m/s2
       
      This procedure gives a set of numbers that are reasonably close to one another: three values in m/s2and three in g. Find the mean of both of these triplets. A partially completed table is provided below. (The intermediate steps are in blue.)

      maximum speed radius centripetal acceleration
      (km/h) (m/s) (m) (m/s2) (g)
          350    
      200 56 705 4.38 0.446
      400 111 2825 4.37 0.446
      500 139 4415 4.37 0.446
      mean centripetal acceleration → 4.37 0.446

    2. For the second part of this question, follow the logic of the first part in reverse order. Assume that the maximum permissible centripetal acceleration is the same for all curves, regardless of size. Use the mean value we just calculated to determine the speed limit on a curve with a 350 m radius.
       
      ac =  v2 ⇒ v = √acr = √(4.372 m/s2)(350 m) = 39.1 m/s
      r
       
      Then convert from m/s to km/h.
       
      39.1 m   1 km   3600 s  = 140 km/h
      1 s 1000 m 1 h
       
      An entirely completed table is provided below. (The intermediate steps are in blue.)

      maximum speed radius centripetal acceleration
      (km/h) (m/s) (m) (m/s2) (g)
      140 39 350 4.37 0.446
      200 56 705 4.38 0.446
      400 111 2,825 4.37 0.446
      500 139 4,415 4.36 0.446
      mean centripetal acceleration → 4.37 0.446

    A more advanced technique is to solve the problem graphically.


    [magnify]
    1. Plot the square of velocity versus radius. This transformation sets up a linear relationship where the slope of the line of best fit will be the centripetal acceleration. Be sure to convert the speeds from km/h to m/s as described above before proceding.
                 
      ac =  v2 ⇒ v2 = acr ⇔ y = mx + b
      r
               
      m  = slope  = ac  = 4.37 m/s2  = 0.445 g
      b  = intercept  = 0  = 7.28 m2/s2  ≈ 0
               
      The intercept value of 7.28 m2/s2is effectively equal to zero. Since the speed squared values are all on the order of several thousand, an intercept that's less than ten is "small" in comparison.
    2. Use the coefficients from the line of best fit to find the speed limit on the minimum radius curve.
       
      y =  mx + b
      v2 =  acr + 0
      v2 =  (4.37 m/s2)(350 m) + (7.28 m2/s2)
      v =  39 m/s = 140 km/h
       
  4. Write something completely different.
    • Answer it.

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.

numerical

  1. A 500 kg race car rounds a curve with a radius of 100 m.
    1. What type of force is 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?
  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 the acceleration provided by gravity compare to the centripetal acceleration?
  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 period of one complete revolution, and
    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

    [magnify]
    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 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 hours. 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. State your answer in terms of
    1. multiples of earth's radius.
    2. the fraction of the distance from the earth to the moon.
    (For now, we will assume the acceleration due to gravity at this distance is negligible. We will not make this assumption once the topic of gravity has been presented in this book.)
  8. Tethered spacecraft.

    [magnify]
    Text

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.

worksheets

  1. These problems are also available on the worksheet circular-motion.pdf.
  2. 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.
    1. Free-body diagrams revisited — I. James E. Court. The Physics Teacher. Vol. 37, No. 7 (October 1999): 427-433. Note: pages 432-433 are relevant to this topic.
    2. Free-body diagrams. James E. Court. The Physics Teacher. Vol. 31, No. 2 (February 1993): 104-108. Note: Questions 20-24 are relevant to this topic.

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