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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.

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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

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(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?

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Questions & Answers

What does mean ohms law imply
Victoria Reply
what is matter
folajin Reply
Anything that occupies space
Kevin
Any thing that has weight and occupies space
Victoria
Anything which we can feel by any of our 5 sense organs
Suraj
Right
Roben
thanks
Suraj
what is a sulphate
Alo
any answers
Alo
the time rate of increase in velocity is called
Blessing Reply
acceleration
Emma
What is uniform velocity
Victoria
Greetings,users of that wonderful app.
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Cruz
P=F/A
Mira
can someone derive the formula a little bit deeper?
Bern
what is coplanar force?
OLADITI Reply
what is accuracy and precision
Peace Reply
How does a current follow?
Vineeta Reply
follow?
akif
which one dc or ac current.
akif
how does a current following?
Vineeta
?
akif
AC current
Vineeta
AC current follows due to changing electric field and magnetic field.
akif
you guys are just saying follow is flow not follow please
Abubakar
ok bro thanks
akif
flows
Abubakar
but i wanted to understand him/her in his own language
akif
but I think the statement is written in English not any other language
Abubakar
my mean that in which form he/she written this,will understand better in this form, i write.
akif
ok
Abubakar
ok thanks bro. my mistake
Vineeta
u are welcome
Abubakar
what is a semiconductor
Vineeta Reply
substances having lower forbidden gap between valence band and conduction band
akif
what is a conductor?
Vineeta
replace lower by higher only
akif
convert 56°c to kelvin
Abubakar
How does a current follow?
Vineeta
A semiconductor is any material whose conduction lies between that of a conductor and an insulator.
AKOWUAH
what is Atom? what is molecules? what is ions?
Abubakar Reply
What is a molecule
Samuel Reply
Is a unit of a compound that has two or more atoms either of the same or different atoms
Justice
A molecule is the smallest indivisible unit of a compound, Just like the atom is the smallest indivisible unit of an element.
Rachel
what is a molecule?
Vineeta
what is a vector
smith Reply
A quantity that has both a magnitude AND a direction. E.g velocity, acceleration, force are all vector quantities. Hope this helps :)
deage
what is the difference between velocity and relative velocity?
Mackson
Velocity is the rate of change of displacement with time. Relative velocity on the other hand is the velocity observed by an observer with respect to a reference point.
Chuks
what do u understand by Ultraviolet catastrophe?
Rufai
A certain freely falling object, released from rest, requires 1.5seconds to travel the last 30metres before it hits the ground. (a) Find the velocity of the object when it is 30metres above the ground.
Mackson
A vector is a quantity that has both magnitude and direction
Rufus
the velocity Is 20m/s-2
Rufus
derivation of electric potential
Rugunda Reply
V = Er = (kq/r^2)×r V = kq/r Where V: electric potential.
Chuks
what is the difference between simple motion and simple harmonic motion ?
syed
hi
Peace
hi
Rufus
hi
Chip
simple harmonic motion is a motion of tro and fro of simple pendulum and the likes while simple motion is a linear motion on a straight line.
Muinat
a body acceleration uniform from rest a 6m/s -2 for 8sec and decelerate uniformly to rest in the next 5sec,the magnitude of the deceleration is ?
Patricia Reply
The wording not very clear kindly
Moses
6
Leo
9.6m/s2
Jolly
the magnitude of deceleration =-9.8ms-2. first find the final velocity using the known acceleration and time. next use the calculated velocity to find the size of deceleration.
Mackson
wrong
Peace
-3.4m/s-2
Justice
Hi
Abj
Firstly, calculate final velocity of the body and then the deceleration. The final ans is,-9.6ms-2
Muinat
8x6= 48m/-2 use v=u + at 48÷5=9.6
Lawrence
can i define motion like this motion can be define as the continuous change of an object or position
Shuaib Reply
Any object in motion will come to rest after a time duration. Different objects may cover equal distance in different time duration. Therefore, motion is defined as a change in position depending on time.
Chuks
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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