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  • Establish the expression for centripetal acceleration.
  • Explain the centrifuge.

We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration.

[link] shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the centripetal acceleration    ( a c size 12{a rSub { size 8{c} } } {} ); centripetal means “toward the center” or “center seeking.”

The given figure shows a circle, with a triangle having vertices A B C made from the center to the boundry. A is at the center and B and C points are at the circle path. Lines A B and A C act as radii and B C is a chord. Delta theta is shown inside the triangle, and the arc length delta s and the chord length delta r are also given. At point B, velocity of object is shown as v one and at point C, velocity of object is shown as v two. Along the circle an equation is shown as delta v equals v sub 2 minus v sub 1.
The directions of the velocity of an object at two different points are shown, and the change in velocity Δ v size 12{Δv} {} is seen to point directly toward the center of curvature. (See small inset.) Because a c = Δ v / Δ t {a rSub { {c} } =Δv/Δt} {} , the acceleration is also toward the center; a c size 12{a rSub { size 8{c} } } {} is called centripetal acceleration. (Because Δ θ size 12{Δθ} {} is very small, the arc length Δ s size 12{Δs} {} is equal to the chord length Δ r size 12{Δr} {} for small time differences.)

The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii r size 12{r} {} and Δ s size 12{Δs} {} are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds v 1 = v 2 = v size 12{v rSub { size 8{1} } =v rSub { size 8{2} } =v} {} . Using the properties of two similar triangles, we obtain

Δ v v = Δ s r . size 12{ { {Δv} over {v} } = { {Δs} over {r} } "."} {}

Acceleration is Δ v / Δ t size 12{Δv/Δt} {} , and so we first solve this expression for Δ v size 12{Δv} {} :

Δ v = v r Δ s . size 12{Δv= { {v} over {r} } Δs"."} {}

Then we divide this by Δ t size 12{Δt} {} , yielding

Δ v Δ t = v r × Δ s Δ t . size 12{ { {Δv} over {Δt} } = { {v} over {r} } times { {Δs} over {Δt} } "."} {}

Finally, noting that Δ v / Δ t = a c size 12{Δv/Δt=a rSub { size 8{c} } } {} and that Δ s / Δ t = v size 12{Δs/Δt=v} {} , the linear or tangential speed, we see that the magnitude of the centripetal acceleration is

a c = v 2 r , size 12{a rSub { size 8{c} } = { {v rSup { size 8{2} } } over {r} } ","} {}

which is the acceleration of an object in a circle of radius r size 12{r} {} at a speed v size 12{v} {} . So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that a c size 12{a rSub { size 8{c} } } {} is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that a c size 12{a rSub { size 8{c} } } {} is greater for tighter turns, as you have probably noticed.

It is also useful to express a c size 12{a rSub { size 8{c} } } {} in terms of angular velocity. Substituting v = size 12{v=rω} {} into the above expression, we find a c = 2 / r = 2 size 12{a rSub { size 8{c} } = left (rω right ) rSup { size 8{2} } /r=rω rSup { size 8{2} } } {} . We can express the magnitude of centripetal acceleration using either of two equations:

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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Kyle
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Adin
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Kyle
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Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, College physics arranged for cpslo phys141. OpenStax CNX. Dec 23, 2014 Download for free at http://legacy.cnx.org/content/col11718/1.4
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