<< Chapter < Page Chapter >> Page >

Weight on an incline, a two-dimensional problem

Consider the skier on a slope shown in [link] . Her mass including equipment is 60.0 kg. (a) What is her acceleration if friction is negligible? (b) What is her acceleration if friction is known to be 45.0 N?

A skier is skiing down the slope and the slope makes a twenty-five degree angle with the horizontal. Her weight W, shown by a vector vertically downward, breaks into two components—one is W parallel, which is shown by a vector arrow parallel to the slope, and the other is W perpendicular, shown by a vector arrow perpendicular to the slope in the downward direction. Vector N is represented by an arrow pointing upward and perpendicular to the slope, having the same length as W perpendicular. Friction vector f is represented by an arrow along the slope in the uphill direction. IIn a free-body diagram, the vector arrow W for weight is acting downward, the vector arrow for f is shown along the direction of the slope, and the vector arrow for N is shown perpendicular to the slope.
Since motion and friction are parallel to the slope, it is most convenient to project all forces onto a coordinate system where one axis is parallel to the slope and the other is perpendicular (axes shown to left of skier). N size 12{N} {} is perpendicular to the slope and f is parallel to the slope, but w size 12{w} {} has components along both axes, namely w size 12{w rSub { size 8{ ortho } } } {} and w . N size 12{N} {} is equal in magnitude to w size 12{w rSub { size 8{ ortho } } } {} , so that there is no motion perpendicular to the slope, but f size 12{f} {} is less than w size 12{w rSub { size 8{ \lline \lline } } } {} , so that there is a downslope acceleration (along the parallel axis).


This is a two-dimensional problem, since the forces on the skier (the system of interest) are not parallel. The approach we have used in two-dimensional kinematics also works very well here. Choose a convenient coordinate system and project the vectors onto its axes, creating two connected one -dimensional problems to solve. The most convenient coordinate system for motion on an incline is one that has one coordinate parallel to the slope and one perpendicular to the slope. (Remember that motions along mutually perpendicular axes are independent.) We use the symbols size 12{ ortho } {} and size 12{ \lline \lline } {} to represent perpendicular and parallel, respectively. This choice of axes simplifies this type of problem, because there is no motion perpendicular to the slope and because friction is always parallel to the surface between two objects. The only external forces acting on the system are the skier’s weight, friction, and the support of the slope, respectively labeled w size 12{w} {} , f size 12{f} {} , and N size 12{N} {} in [link] . N size 12{N} {} is always perpendicular to the slope, and f size 12{f} {} is parallel to it. But w size 12{w} {} is not in the direction of either axis, and so the first step we take is to project it into components along the chosen axes, defining w size 12{w rSub { size 8{ \lline \lline } } } {} to be the component of weight parallel to the slope and w size 12{w rSub { size 8{ ortho } } } {} the component of weight perpendicular to the slope. Once this is done, we can consider the two separate problems of forces parallel to the slope and forces perpendicular to the slope.


The magnitude of the component of the weight parallel to the slope is w = w sin ( 25º ) = mg sin ( 25º ) size 12{w rSub { size 8{ \lline \lline } } =w"sin" \( "25"° \) = ital "mg""sin" \( "25"° \) } {} , and the magnitude of the component of the weight perpendicular to the slope is w = w cos ( 25º ) = mg cos ( 25º ) size 12{w rSub { size 8{ ortho } } =w"cos" \( "25"° \) = ital "mg""cos" \( "25"° \) } {} .

(a) Neglecting friction. Since the acceleration is parallel to the slope, we need only consider forces parallel to the slope. (Forces perpendicular to the slope add to zero, since there is no acceleration in that direction.) The forces parallel to the slope are the amount of the skier’s weight parallel to the slope w size 12{w rSub { size 8{ \lline \lline } } } {} and friction f size 12{f} {} . Using Newton’s second law, with subscripts to denote quantities parallel to the slope,

a = F net m

where F net = w = mg sin ( 25º ) size 12{F rSub { size 8{"net " \lline \lline } } =w rSub { size 8{ \lline \lline } } = ital "mg""sin" \( "25"° \) } {} , assuming no friction for this part, so that

a = F net m = mg sin ( 25º ) m = g sin ( 25º )
( 9.80 m/s 2 ) ( 0.4226 ) = 4.14 m/s 2

is the acceleration.

(b) Including friction. We now have a given value for friction, and we know its direction is parallel to the slope and it opposes motion between surfaces in contact. So the net external force is now

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics arranged for cpslo phys141' conversation and receive update notifications?