A person standing on the ground needs to lift a 100.0kg box to a ledge that is 2.000m high. He sets up a pulley
system to do this. He pulls down on the effort rope of the pulley system with a force of 245.0N, and has to pull
10.00m of rope to lift the box to the 2.000m ledge. Answer the following questions.
1)
Calculate the output force.
݉ = 100.0݇݃
݃ = 9.81
2)
௠
௦మ
‫ܨ‬௢௨௧ = ‫݃݅݁ݓ‬ℎ‫݃݉ = ݐ‬
௠
‫ܨ‬௢௨௧ = (100݇݃)(9.81 మ )
௦
10m
‫ܨ‬௢௨௧ = 981ܰ
2m
Calculate the MA of the pulley system used.
‫ܨ‬௜௡ = 245ܰ
‫= ܣܯ‬
ி೚ೠ೟
ி೔೙
‫ = ܣܯ‬4.00
=
ଽ଼ଵே
ଶସହே
3)
Calculate the work input for this machine.
݀௜௡ = 10.00݉
ܹ௜௡ = ‫ܨ‬௜௡ ݀௜௡
ܹ௜௡ = ሺ245ܰሻሺ10݉ሻ = 2450‫ܬ‬
ܹ௜௡ = 2.45 × 10ଷ ‫ܬ‬
4)
Calculate the work output for this machine.
ܹ௢௨௧ = ‫ܨ‬௢௨௧ ݀௢௨௧
݀௢௨௧ = 2.00݉
ܹ௢௨௧ = ሺ981ܰሻሺ2݉ሻ = 1962‫ܬ‬
ܹ௢௨௧ = 1.96 × 10ଷ ‫ܬ‬
5)
Calculate the efficiency for this machine.
ௐ
݂݂݁ = ௐ೚ೠ೟ × 100
݂݂݁ =
೔೙
ଵଽ଺ଶ௃
ଶସହ଴௃
× 100
݂݂݁ = 80.1%
6)
Why wasn’t this machine 100% efficient?
Friction
7)
Tell all of the ways that this pulley system changed the force, and explain, using data from these
problems, how you know. It multiplied the input force by increasing the distance the input force acted
over and changed the direction of force
MACHINE PROBLEMS 2
Name
A 12.5kg mass is pulled up a 15.0m long inclined plane at a constant velocity by an effort of 22.0N. If the inclined
plane has a height of 2.50m, calculate the following.
1) IMA of the plane
݀௜௡ = 15.0݉
݀௢௨௧ = 2.50݉
݀௜௡
15݉
=
݀௢௨௧ 2.5݉
‫ = ܣܯܫ‬6
‫= ܣܯܫ‬
2) Output force
2.5m
ߠ
‫ܨ‬௢௨௧ = ‫݃݅݁ݓ‬ℎ‫݃݉ = ݐ‬
௠
‫ܨ‬௢௨௧ = (12.5݇݃)(9.81 ௦మ )
݉ = 12.5݇݃
݃ = 9.81 ௦మ
௠
3) MA of the plane
‫ܨ‬௜௡ = 22.0ܰ
‫= ܣܯ‬
‫ܨ‬௢௨௧ = 122.6ܰ
‫ܨ‬௢௨௧ = 123ܰ
ி೚ೠ೟
ி೔೙
=
‫ = ܣܯ‬5.57
4) Work input
݀௜௡ = 15݉
5) Work output
݀௢௨௧ = 2.50݉
ଵଶଶ.଺ே
ଶଶே
ܹ௜௡ = ‫ܨ‬௜௡ ݀௜௡
ܹ௜௡ = ሺ22ܰሻሺ15݉ሻ = 330‫ܬ‬
ܹ௜௡ = 3.30 × 10ଶ ‫ܬ‬
ܹ௢௨௧ = ‫ܨ‬௢௨௧ ݀௢௨௧
ܹ௢௨௧ = ሺ122.6ܰሻሺ2.5݉ሻ = 306.5‫ܬ‬
ܹ௢௨௧ = 307‫ܬ‬
ܹ௢௨௧
× 100
ܹ௜௡
ଷ଴଺.ହ௃
݂݂݁ = ଷଷ଴௃ × 100
6) Efficiency
݂݂݁ =
݂݂݁ = 92.9%
7) Coefficient of friction between mass and plane. (Hint: 5 steps.)
‫ܨ‬௙ = ߤ‫ܨ‬ே
Find Force Of Friction
By conservation of energy
ܹ௜௡ = ܹ௢௨௧ + ܹ௙௥௜௖௧௜௢௡
݀௙ = ݀௙௥௜௖௧௜௢௡ = 15݉
330‫ = ܬ‬306.5‫ ܬ‬+ ‫ܨ‬௙ ݀௙
330‫ = ܬ‬306.5‫ ܬ‬+ ‫ܨ‬௙ ሺ15݉ሻ
‫ܨ‬௙ = 1.576ܰ
From Drawing
‫ܨ‬ே = ‫ܨ‬௚ୄ = ݉݃ܿ‫ߠݏ݋‬
Find Normal Force
Find ߤ
ߤ = ி೑ =
ி
ಿ
ଵ.ହ଻଺ே
ߤ = .013
ଵଶ଴.ଽ
ߠ = ‫ି݊݅ݏ‬ଵ ቀ ଵହ௠ ቁ = 9.59௢
ଶ.ହ௠
‫ܨ‬ே = ሺ12.5݇݃ሻ ቀ9.81 ௦మ ቁ cosሺ9.59ሻ = 120.9ܰ
௠
SM 1. The efficiency of a squeaky pulley system is 73 %. The pulleys are used to raise a mass. What
force is exerted on the machine if a rope is pulled 18.0 m in order to raise a 58 kg mass a height of 3.0 m?
݂݂݁ = 73%
݂݂݁ =
‫ܨ‬௜௡ =?
݂݂݁ =
݀௜௡ = 18.0݉
ௐ೚ೠ೟
ௐ೔೙
ி೚ೠ೟ ௗ೚ೠ೟
73% =
݉ = 58݇݃
× 100
݀௢௨௧ = 3.0݉
ி೔೙ ௗ೔೙
× 100
(ହ଺ଽே)(ଷ௠)
(ி೔೙ )(ଵ଼௠)
× 100
‫ܨ‬௜௡ = 129.9ܰ
‫ܨ‬௜௡ = 1.3 × 10ଶ ܰ
݉
‫ܨ‬௢௨௧ = ݉݃ = ሺ58݇݃ሻ ቀ9.91 ଶ ቁ = 569ܰ
‫ݏ‬
SM 2. The ramp in this figure is 18 m long and 4.5 m high.
a. What force parallel to the ramp (FA) is required to slide a 25 kg box at constant speed to the top of the
ramp if friction is disregarded?
‫ܨ‬௚∥ = ‫ܨ‬௚ ‫ߠ݊݅ݏ݃݉ = ߠ݊݅ݏ‬
݉ = 25݇݃
‫ܨ‬௚∥ = (25݇݃)(9.81 ௦మ )sin(14.5௢ )
Must push against
݃ = 9.81 ௦మ
௠
‫ܨ‬௚∥ = 61.4ܰ
௠
4.5
ߠ = ‫ି݊݅ݏ‬ଵ ൬ ൰ = 14.5௢
18
b. What is the IMA of the ramp?
݀௜௡ = 18݉
‫ = ܣܯܫ‬ௗ ೔೙
ௗ
‫ = ܣܯܫ‬ସ.ହ௠
೚ೠ೟
ଵ଼௠
݀௢௨௧ = 4.5݉
‫ = ܣܯܫ‬4
c. What are the real MA and the efficiency of the ramp if a parallel force (FA) of 75 N is actually
ி
ଶସହ.ଶହே
required?
‫ = ܣܯ‬ி೚ೠ೟ = ଻ହே = 3.27 = ‫ܣܯ‬
‫ܨ‬௢௨௧ = ݉݃ = ሺ25݇݃ሻ ቀ9.81 ௦మ ቁ = 245.25ܰ
೔೙
௠
݂݂݁ =
‫ܣܯ‬
3.27
× 100 =
× 100 = 81.8%
‫ܣܯܫ‬
4
SM 3. A person must move a large stone in her backyard. She gets a plank that is 3.50 m long to use as a
lever and she wedges one end of the plank under the large stone. A smaller stone, 0.750 m from the large
one, serves as a pivot. The plank makes an angle of 20.0 o with the ground. When she pushes down on
the end of the plank with a force of 210 N, the large stone begins to move.
‫ܨ‬௜௡ = 210ܰ
݀௜௡ = 2.75݉
݀௢௨௧ = 0.75݉
You can use the lever arms as distances
‫ܨ‬௢௨௧ =?
ܹ௜௡ ≥ ܹ௢௨௧
‫ܨ‬௜௡ ݀௜௡ ≥ ‫ܨ‬௢௨௧ ݀௢௨௧
ሺ210ܰሻሺ2.75݉ሻ = ሺ‫ܨ‬௢௨௧ ሻ(0.75݉)
a. What upward force is being exerted against the large stone?
‫ܨ‬௢௨௧ = 770ܰ
b. What is the mechanical advantage of the lever?
‫= ܣܯ‬
ி೚ೠ೟
ி೔೙
‫ܨ‬௢௨௧ = 7.70 × 10ଶ ܰ
଻଻଴ே
= ଶଵ଴ே = 3.67
SM 4. Why is it easier to loosen the lid from the top of a paint can with a long handled screwdriver than
with a short-handled screwdriver?
More Leverage: Longer handle means more input distance so there is more output force
SM 5. You are attempting to move a large rock using a long lever. Is it more effective to place the
lever’s axis of rotation nearer to your hands or nearer to the rock? Explain.
It is more effective to put the lever’s axis (fulcrum) nearer the rock so that your input distance is
greater than output distance thereby making the output force bigger than the input force.
SM 6. A pulley system lifts a 1345 N weight a distance of 0.975 m. Paul pulls the rope a distance of 3.90
m, exerting a force of 375 N.
a. What is the IMA of the system?
‫ܨ‬௢௨௧ = 1345ܰ
‫ = ܣܯܫ‬ௗ ೔೙ = ଴.ଽ଻ହ௠
b. What is the MA?
‫= ܣܯ‬
݀௢௨௧ = 0.975݉
݀௜௡ = 3.90݉
‫ܨ‬௜௡ = 375ܰ
c. How efficient is the system?
ௗ
ଷ.ଽ௠
೚ೠ೟
‫ = ܣܯܫ‬4
ி೚ೠ೟
ி೔೙
=
ଵଷସହே
ଷ଻ହே
= 3.587
݂݂݁ = ூெ஺ × 100 =
ெ஺
ଷ.ହ଼଻
ସ
‫ = ܣܯ‬3.59
× 100 = 89.7%
SM 7. If you were to use a machine to increase the output force, what factor would have to be sacrificed?
Give an example where this occurs. You have to sacrifice the distance the output force moves
compared to the distance the input moves.
SM 8. A leveling foot on an air track is a screw. It can be turned to move it into or out of the base of the
air track to make the end go up or down. Suppose the leveling screw on a particular air track has 15
threads per cm. The screw has a knob of diameter 1.5 cm to make it easier to turn.
1ܿ݉ ‫ = ݌ݑ‬15 ‫ݐ ݂݋ ݏ݊݋݅ݐݑ݈݋ݒ݁ݎ‬ℎ݁ ‫ݐ‬ℎ‫݀ܽ݁ݎ‬
a. In one complete rotation of the screw, how far is the air track lifted?
Therefore
ଵ ௥௘௩௢௟௨௧௜௢௡
ଵ
ቀଵହ ௥௘௩௢௟௨௧௜௢௡௦ቁ = 0.0667ܿ݉ ‫݌ݑ‬
ଵ ௖௠ ௨௣
= ݀௢௨௧
b. What is the IMA of the screw?
݀௜௡ = ܿ݅‫ = ߨ݀ = ݁ܿ݊݁ݎ݂݁݉ݑܿݎ‬ሺ1.5ܿ݉ሻߨ = 4.71ܿ݉
‫ = ܣܯܫ‬ௗ ೔೙ = ଴.଴଺଺଻௖௠ = 70.69
ௗ
ସ.଻ଵ௖௠
೚ೠ೟
‫ = ܣܯܫ‬70.7
c. If a force of 0.50 N is necessary to rotate the knob to lift the air track, what is the weight of the end of
the air track lifted, assuming the screw is 80% efficient?
Weight = output force
Need to know MA
݂݂݁ = 80%
‫ܨ‬௜௡ = 0.50ܰ
ெ஺
݂݂݁ = ூெ஺ × 100
ெ஺
80% = ଻଴.଻ × 100
‫ = ܣܯ‬56.5
ி೚ೠ೟
ி೔೙
ி೚ೠ೟
= ଴.ହே
‫= ܣܯ‬
56.5
‫ܨ‬௢௨௧ = 28.3ܰ