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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

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Unit 4 - uniform circular motion and universal law of gravity. OpenStax CNX. Nov 23, 2015 Download for free at https://legacy.cnx.org/content/col11905/1.1
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