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F net = w f size 12{F rSub { size 8{"net " \lline \lline } } =w rSub { size 8{ \lline \lline } } - f} {} ,

and substituting this into Newton’s second law, a = F net m size 12{a rSub { size 8{ \lline \lline } } = { {F rSub { size 8{"net " \lline \lline } } } over {m} } } {} , gives

a = F net m = w f m = mg sin ( 25º ) f m size 12{a rSub { size 8{ \lline \lline } } = { {F rSub { size 8{"net " \lline \lline } } } over {m} } = { {w rSub { size 8{ \lline \lline } } - f} over {m} } = { { ital "mg""sin" \( "25"° \) - f} over {m} } } {} .

We substitute known values to obtain

a = ( 60 . 0 kg ) ( 9 . 80 m/s 2 ) ( 0 . 4226 ) 45 . 0 N 60 . 0 kg size 12{a rSub { size 8{ \lline \lline } } = { { \( "60" "." "0 kg" \) \( 9 "." "80 m/s" rSup { size 8{2} } \) \( 0 "." "4226" \) - "45" "." "0 N"} over {"60" "." "0 kg"} } } {} ,

which yields

a = 3 . 39 m/s 2 size 12{a rSub { size 8{ \lline \lline } } =3 "." "39 m/s" rSup { size 8{2} } } {} ,

which is the acceleration parallel to the incline when there is 45.0 N of opposing friction.


Since friction always opposes motion between surfaces, the acceleration is smaller when there is friction than when there is none. In fact, it is a general result that if friction on an incline is negligible, then the acceleration down the incline is a = g sin θ size 12{a=g"sin"θ} {} , regardless of mass . This is related to the previously discussed fact that all objects fall with the same acceleration in the absence of air resistance. Similarly, all objects, regardless of mass, slide down a frictionless incline with the same acceleration (if the angle is the same).

Resolving weight into components

Vector arrow W for weight is acting downward. It is resolved into components that are parallel and perpendicular to a surface that has a slope at angle theta to the horizontal. The coordinate direction x is labeled parallel to the sloped surface, with positive x pointing uphill. The coordinate direction y is labeled perpendicular to the sloped surface, with positive y pointing up from the surface. The components of w are w parallel, represented by an arrow pointing downhill along the sloped surface, and w perpendicular, represented by an arrow pointing into the sloped surface. W parallel is equal to w sine theta, which is equal to m g sine theta. W perpendicular is equal to w cosine theta, which is equal to m g cosine theta.
An object rests on an incline that makes an angle θ with the horizontal.

When an object rests on an incline that makes an angle θ size 12{θ} {} with the horizontal, the force of gravity acting on the object is divided into two components: a force acting perpendicular to the plane, w size 12{w rSub { size 8{ ortho } } } {} , and a force acting parallel to the plane, w size 12{w rSub { size 8{ \lline \lline } } } {} . The perpendicular force of weight, w size 12{w rSub { size 8{ ortho } } } {} , is typically equal in magnitude and opposite in direction to the normal force, N size 12{N} {} . The force acting parallel to the plane, w size 12{w rSub { size 8{ \lline \lline } } } {} , causes the object to accelerate down the incline. The force of friction, f size 12{f} {} , opposes the motion of the object, so it acts upward along the plane.

It is important to be careful when resolving the weight of the object into components. If the angle of the incline is at an angle θ size 12{θ} {} to the horizontal, then the magnitudes of the weight components are

w = w sin ( θ ) = mg sin ( θ ) size 12{w rSub { size 8{ \lline \lline } } =w"sin" \( θ \) = ital "mg""sin" \( θ \) " "} {}


w = w cos ( θ ) = mg cos ( θ ) size 12{w rSub { size 8{ ortho } } =w"cos" \( θ \) = ital "mg""cos" \( θ \) } {} .

Instead of memorizing these equations, it is helpful to be able to determine them from reason. To do this, draw the right triangle formed by the three weight vectors. Notice that the angle θ size 12{θ} {} of the incline is the same as the angle formed between w size 12{w} {} and w size 12{w rSub { size 8{ ortho } } } {} . Knowing this property, you can use trigonometry to determine the magnitude of the weight components:

cos ( θ ) = w w w = w cos ( θ ) = mg cos ( θ ) alignl { stack { size 12{"cos" \( θ \) = { {w rSub { size 8{ ortho } } } over {w} } } {} #w rSub { size 8{ ortho } } =w"cos" \( θ \) = ital "mg""cos" \( θ \) {} } } {}

sin ( θ ) = w w w = w sin ( θ ) = mg sin ( θ ) alignl { stack { size 12{"sin" \( θ \) = { {w rSub { size 8{ \lline \lline } } } over {w} } } {} #w rSub { size 8{ \lline \lline } } =w"sin" \( θ \) = ital "mg""sin" \( θ \) {} } } {}

Take-home experiment: force parallel

To investigate how a force parallel to an inclined plane changes, find a rubber band, some objects to hang from the end of the rubber band, and a board you can position at different angles. How much does the rubber band stretch when you hang the object from the end of the board? Now place the board at an angle so that the object slides off when placed on the board. How much does the rubber band extend if it is lined up parallel to the board and used to hold the object stationary on the board? Try two more angles. What does this show?


A tension     is a force along the length of a medium, especially a force carried by a flexible medium, such as a rope or cable. The word “tension comes from a Latin word meaning “to stretch.” Not coincidentally, the flexible cords that carry muscle forces to other parts of the body are called tendons . Any flexible connector, such as a string, rope, chain, wire, or cable, can exert pulls only parallel to its length; thus, a force carried by a flexible connector is a tension with direction parallel to the connector. It is important to understand that tension is a pull in a connector. In contrast, consider the phrase: “You can’t push a rope.” The tension force pulls outward along the two ends of a rope.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
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I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
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what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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ramon Reply
Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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what is system testing
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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silver nanoparticles could handle the job?
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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