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Calculating distance traveled: sliding up an incline

Suppose that the player from [link] is running up a hill having a 5 . 00º size 12{5 "." "00"°} {} incline upward with a surface similar to that in the baseball stadium. The player slides with the same initial speed. Determine how far he slides.

A baseball player slides on an inclined slope represented by a right triangle. The angle of the slope is represented by the angle between the base and the hypotenuse, which is equal to five degrees, and the height h of the perpendicular side of the triangle is equal to d sin 5 degrees. The length of the hypotenuse is d.
The same baseball player slides to a stop on a 5.00º size 12{5°} {} slope.

Strategy

In this case, the work done by the nonconservative friction force on the player reduces the mechanical energy he has from his kinetic energy at zero height, to the final mechanical energy he has by moving through distance d size 12{d} {} to reach height h size 12{h} {} along the hill, with h = d sin 5.00º size 12{h=d"sin"5 "." "00"°} {} . This is expressed by the equation

KE i + PE i + W nc = KE f + PE f . size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE""" lSub { size 8{f} } +"PE" rSub { size 8{f} } } {}

Solution

The work done by friction is again W nc = fd size 12{W rSub { size 8{"nc"} } = - ital "fd"} {} ; initially the potential energy is PE i = mg 0 = 0 and the kinetic energy is KE i = 1 2 mv i 2 ; the final energy contributions are KE f = 0 size 12{"KE" rSub { size 8{f} } =0} {} for the kinetic energy and PE f = mgh = mgd sin θ size 12{"PE" rSub { size 8{f} } = ital "mgh"= ital "mgd""sin"θ} {} for the potential energy.

Substituting these values gives

1 2 mv i 2 + 0 + ( fd ) = 0 + mgd sin θ.

Solve this for d to obtain

d = 1 2 mv i 2 f + mg sin θ = (0.5) ( 65.0 kg ) ( 6.00 m/s ) 2 450 N + ( 65.0 kg ) ( 9.80 m/s 2 ) sin (5.00º) = 2.31 m. alignl { stack { size 12{d = { { left ( { {1} over {2} } right ) ital "mv""" lSub { size 8{i} } "" lSup { size 8{2} } } over {f+ ital "mg""sin5" rSup { size 8{ circ } } } } } {} #" "= { {0 "." "5 " \( "65" "." "0 kg" \) \( 6 "." "00 m/s" \) rSup { size 8{2} } } over {"450""N "+ \( "65" "." "0 kg" \) \( 9 "." "80 m/s" rSup { size 8{2} } \) "sin 5" rSup { size 8{ circ } } } } {} # " "=2 "." "31 m" "." {}} } {}

Discussion

As might have been expected, the player slides a shorter distance by sliding uphill. Note that the problem could also have been solved in terms of the forces directly and the work energy theorem, instead of using the potential energy. This method would have required combining the normal force and force of gravity vectors, which no longer cancel each other because they point in different directions, and friction, to find the net force. You could then use the net force and the net work to find the distance d size 12{d} {} that reduces the kinetic energy to zero. By applying conservation of energy and using the potential energy instead, we need only consider the gravitational potential energy mgh size 12{ ital "mgh"} {} , without combining and resolving force vectors. This simplifies the solution considerably.

Making connections: take-home investigation—determining friction from the stopping distance

This experiment involves the conversion of gravitational potential energy into thermal energy. Use the ruler, book, and marble from Take-Home Investigation—Converting Potential to Kinetic Energy . In addition, you will need a foam cup with a small hole in the side, as shown in [link] . From the 10-cm position on the ruler, let the marble roll into the cup positioned at the bottom of the ruler. Measure the distance d size 12{d} {} the cup moves before stopping. What forces caused it to stop? What happened to the kinetic energy of the marble at the bottom of the ruler? Next, place the marble at the 20-cm and the 30-cm positions and again measure the distance the cup moves after the marble enters it. Plot the distance the cup moves versus the initial marble position on the ruler. Is this relationship linear?

With some simple assumptions, you can use these data to find the coefficient of kinetic friction μ k size 12{μ rSub { size 8{k} } } {} of the cup on the table. The force of friction f on the cup is μ k N size 12{μ rSub { size 8{k} } N} {} , where the normal force N is just the weight of the cup plus the marble. The normal force and force of gravity do no work because they are perpendicular to the displacement of the cup, which moves horizontally. The work done by friction is fd size 12{ ital "fd"} {} . You will need the mass of the marble as well to calculate its initial kinetic energy.

It is interesting to do the above experiment also with a steel marble (or ball bearing). Releasing it from the same positions on the ruler as you did with the glass marble, is the velocity of this steel marble the same as the velocity of the marble at the bottom of the ruler? Is the distance the cup moves proportional to the mass of the steel and glass marbles?

A marble is rolling down a makeshift ramp consisting of a small wooden ruler propped up on one end at about a thirty degree angle. At the bottom of the ramp is a foam drinking cup standing upside-down on its lip. A hole is cut out on one side of the cup so that the marble will roll through the hole when it reaches the bottom of the ramp.
Rolling a marble down a ruler into a foam cup.

Phet explorations: the ramp

Explore forces, energy and work as you push household objects up and down a ramp. Lower and raise the ramp to see how the angle of inclination affects the parallel forces acting on the file cabinet. Graphs show forces, energy and work.

The Ramp

Section summary

  • A nonconservative force is one for which work depends on the path.
  • Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
  • Work W nc size 12{W rSub { size 8{"nc"} } } {} done by a nonconservative force changes the mechanical energy of a system. In equation form, W nc = Δ KE + Δ PE size 12{W rSub { size 8{"nc"} } =Δ"KE"+Δ"PE"} {} or, equivalently, KE i + PE i + W nc = KE f + PE f size 12{"KE" rSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } } {} .
  • When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton’s laws.

Problems&Exercises

A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m-high rise as shown in [link] . Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.)

A skier is about to go up an inclined slope with some initial speed v sub i shown by an arrow towards right. The slope makes a thirty-five-degree with the horizontal. The height of the point where the slope ends from the skiers’ starting position is two point five meters. Final speed of the skier at the end of the inclined slope is unknown.
The skier’s initial kinetic energy is partially used in coasting to the top of a rise.

9.46 m/s

(a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope 2 . size 12{2 "." 5°} {} above the horizontal?

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Une: physics for the health professions. OpenStax CNX. Aug 20, 2014 Download for free at http://legacy.cnx.org/content/col11697/1.1
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