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f k = μ k mg cos 25º , size 12{f rSub { size 8{k} } =μ rSub { size 8{k} } ital "mg""cos""25" rSup { size 8{ circ } } } {}

which can now be solved for the coefficient of kinetic friction μ k size 12{μ rSub { size 8{k} } } {} .

Solution

Solving for μ k size 12{μ rSub { size 8{k} } } {} gives

μ k = f k N = f k w cos 25º = f k mg cos 25º . size 12{μ rSub { size 8{k} } = { {f rSub { size 8{k} } } over {N} } = { {f rSub { size 8{k} } } over {w"cos""25" rSup { size 8{ circ } } } } = { {f rSub { size 8{k} } } over { ital "mg""cos""25" rSup { size 8{ circ } } } } } {}

Substituting known values on the right-hand side of the equation,

μ k = 45.0 N ( 62 kg ) ( 9 . 80 m /s 2 ) ( 0 . 906 ) = 0 . 082 . size 12{μ rSub { size 8{k} } = { {"45" "." 0N} over { \( "60"`"kg" \) \( 9 "." "80"`"m/s" rSup { size 8{2} } \) \( 0 "." "906" \) } } =0 "." "084"} {}

Discussion

This result is a little smaller than the coefficient listed in [link] for waxed wood on snow, but it is still reasonable since values of the coefficients of friction can vary greatly. In situations like this, where an object of mass m slides down a slope that makes an angle θ size 12{θ} {} with the horizontal, friction is given by f k = μ k mg cos θ size 12{f rSub { size 8{k} } =μ rSub { size 8{k} } ital "mg""cos"θ} {} . All objects will slide down a slope with constant acceleration under these circumstances. Proof of this is left for this chapter’s Problems and Exercises.

Take-home experiment

An object will slide down an inclined plane at a constant velocity if the net force on the object is zero. We can use this fact to measure the coefficient of kinetic friction between two objects. As shown in [link] , the kinetic friction on a slope f k = μ k mg cos θ size 12{f rSub { size 8{k} } =μ rSub { size 8{k} } ital "mg""cos"θ} {} . The component of the weight down the slope is equal to mg sin θ size 12{ ital "mg""sin"θ} {} (see the free-body diagram in [link] ). These forces act in opposite directions, so when they have equal magnitude, the acceleration is zero. Writing these out:

f k = Fg x size 12{f rSub { size 8{k} } = ital "Fgx"} {}
μ k mg cos θ = mg sin θ . size 12{μ rSub { size 8{k} } ital "mg""cos"θ= ital "mg""sin"θ} {}

Solving for μ k size 12{μ rSub { size 8{k} } } {} , we find that

μ k = mg sin θ mg cos θ = tan θ . size 12{μ rSub { size 8{k} } = { { ital "mg""sin"θ} over { ital "mg""cos"θ} } ="tan"θ} {}

Put a coin on a book and tilt it until the coin slides at a constant velocity down the book. You might need to tap the book lightly to get the coin to move. Measure the angle of tilt relative to the horizontal and find μ k size 12{μ rSub { size 8{K} } } {} . Note that the coin will not start to slide at all until an angle greater than θ size 12{θ} {} is attained, since the coefficient of static friction is larger than the coefficient of kinetic friction. Discuss how this may affect the value for μ k size 12{μ rSub { size 8{K} } } {} and its uncertainty.

We have discussed that when an object rests on a horizontal surface, there is a normal force supporting it equal in magnitude to its weight. Furthermore, simple friction is always proportional to the normal force.

Making connections: submicroscopic explanations of friction

The simpler aspects of friction dealt with so far are its macroscopic (large-scale) characteristics. Great strides have been made in the atomic-scale explanation of friction during the past several decades. Researchers are finding that the atomic nature of friction seems to have several fundamental characteristics. These characteristics not only explain some of the simpler aspects of friction—they also hold the potential for the development of nearly friction-free environments that could save hundreds of billions of dollars in energy which is currently being converted (unnecessarily) to heat.

[link] illustrates one macroscopic characteristic of friction that is explained by microscopic (small-scale) research. We have noted that friction is proportional to the normal force, but not to the area in contact, a somewhat counterintuitive notion. When two rough surfaces are in contact, the actual contact area is a tiny fraction of the total area since only high spots touch. When a greater normal force is exerted, the actual contact area increases, and it is found that the friction is proportional to this area.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Newton's laws. OpenStax CNX. Oct 25, 2015 Download for free at https://legacy.cnx.org/content/col11898/1.1
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