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6.1 Centripetal acceleration  (Page 2/5)

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A centrifuge (see [link] b) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity ( g ) size 12{g} {} ; maximum centripetal acceleration of several hundred thousand g is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth’s gravity.

How does the centripetal acceleration of a car around a curve compare with that due to gravity?

What is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m/s (about 90 km/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See [link] (a).

Strategy

Because v size 12{v} {} and r size 12{r} {} are given, the expression a c = v 2 r size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } `; a rSub { size 8{c} } =rω rSup { size 8{2} } } {} isconvenient to use.

Solution

Entering the given values of v = 25 . 0 m/s size 12{v="25" "." 0`"m/s"} {} and r = 500 m size 12{r="500"} {} into the first expression for a c size 12{a rSub { size 8{c} } } {} gives

a c = v 2 r = ( 25 . 0 m/s ) 2 500 m = 1 . 25 m/s 2 . size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } = { { \( "25" "." 0" m/s" \) rSup { size 8{2} } } over {"500 m"} } =1 "." "25"" m/s" rSup { size 8{2} } "."} {}

Discussion

To compare this with the acceleration due to gravity ( g = 9 . 80 m/s 2 ) size 12{g=9 "." 8`"m/s" rSup { size 8{2} } } {} , we take the ratio of a c / g = 1 . 25 m/s 2 / 9 . 80 m/s 2 = 0 . 128 size 12{a rSub { size 8{c} } /g= left (1 "." "25"`"m/s" rSup { size 8{2} } right )/ left (9 "." "80"`"m/s" rSup { size 8{2} } right )=0 "." "128"} {} . Thus, a c = 0 . 128 g size 12{a rSub { size 8{c} } =0 "." "128"} {} and is noticeable especially if you were not wearing a seat belt.

In figure a, a car shown from top is running on a circular road around a circular path. The center of the park is termed as the center of this circle and the distance from this point to the car is taken as radius r. The linear velocity is shown in perpendicular direction toward the front of the car, shown as v the centripetal acceleration is shown with an arrow pointed towards the center of rotation. In figure b, a centrifuge is shown an object of mass m is rotating in it at a constant speed. The object is at the distance equal to the radius, r, of the centrifuge. The centripetal acceleration is shown towards the center of rotation, and the velocity, v is shown perpendicular to the object in the clockwise direction.
(a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in [link] . (b) A particle of mass in a centrifuge is rotating at constant angular velocity . It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in [link] .

Section summary

  • Centripetal acceleration a c size 12{a rSub { size 8{c} } } {} is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity v size 12{v} {} and has the magnitude
    a c = v 2 r . size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } `; a rSub { size 8{c} } =rω rSup { size 8{2} } } {}
  • The unit of centripetal acceleration is m / s 2 size 12{m/s rSup { size 8{2} } } {} .

Conceptual questions

Can centripetal acceleration change the speed of circular motion? Explain.

Problem exercises

A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?

12.9 rev/min

A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If he completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track?
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Source:  OpenStax, Introductory physics - for kpu phys 1100 (2015 edition). OpenStax CNX. May 30, 2015 Download for free at http://legacy.cnx.org/content/col11588/1.13
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