Download Work Problems Mr. Kepple

Name: _______________________
Mr. Kepple
Work Problems
Work and Energy – HW#1
Date: ___________ Period: _____
1. How much work is done by the gravitational
force when a 280-kg pile driver falls 2.80 m?
2. A 75.0-kg firefighter climbs a flight of stairs
20.0 m high. How much work is required?
𝑊𝐺 = 𝐹𝐺 𝑑 cos 𝜃
𝑊climb = 𝐹climb 𝑑 cos 𝜃
𝑊 = (𝑚𝑔)𝑑 cos 𝜃
𝑊 = (𝑚𝑔)𝑑 cos 𝜃
𝑊 = (280)(9.8)(2.80) cos(0)
𝑊 = (75)(9.8)(20.0) cos(0)
𝑊 = 7683.2 J
𝑊 = 14700 J
𝑊 ≈ 7.7 × 103 J
𝑊 = 1.47 × 104 J
3. What is the minimum work needed to push a 950-kg car 310 m up along a 9.0° incline? Ignore
friction. Make sure you draw a free body diagram!
+𝑦
𝑁
+𝑥
𝐹𝑃
The minimum work is when
the car has constant velocity.
Σ𝐹 = 0
𝜃
𝑚𝑔
𝐹𝑃 − 𝑚𝑔 sin 𝜃 = 0
𝐹𝑃 = 𝑚𝑔 sin 𝜃
𝑊𝑃 = 𝐹𝑃 𝑑 cos 0°
𝑊𝑃 = (𝑚𝑔 sin 𝜃) 𝑑
𝑊𝑃 = (950)(9.80)(310) sin 9.0°
𝑊𝑃 = 451485.51 J
𝑊𝑃 ≈ 4.5 × 105 J
4. A box of mass 6.0 kg is accelerated from rest by a force across a floor at a rate of 2.0 m/s² for 7.0 s.
Find the net work done on the box.
1
𝑑 = 𝑣𝑜 𝑡 + 𝑎𝑡 2
2
1
𝑑 = 0 + 𝑎𝑡 2
2
1
𝑑 = 𝑎𝑡 2
2
𝑊 = 𝐹𝑑 cos 𝜃
1
𝑊 = (𝑚𝑎)( 𝑎𝑡 2 ) cos 0°
2
1
𝑊 = 𝑚𝑎2 𝑡 2
2
1
𝑊 = (6.0)(2.0)2 (7.0)2
2
𝑊 = 588 J
𝑊 ≈ 590 J
5. A 17,000-kg jet takes off from an aircraft carrier
via a catapult. The gases thrust out from the jet’s
engines exert a constant force of 130 kN on the jet;
the force exerted on the jet by the catapult is
plotted in the figure. Determine: (a) The work done
on the jet by the gases expelled by its engines
during launch of the jet; and (b) the work done on
the jet by the catapult during launch of the jet.
(a) Work done by gases
(b) Work done by catapult
𝑊 = 𝐹𝑑 cos 𝜃
1
𝑊 = (1100 × 103 + 65 × 103 )(85)
2
𝑊 = (130 × 103 )(85) cos 0°
𝑊 = 49512500 J ≈ 5.0 × 107 J
𝑊 = 11050000 ≈ 1.1 × 107 J
6. A 2200-N crate rests on the floor. How much work is required to move it at constant speed (a) 4.0 m
along the floor against a drag force of 230 N, and (b) 4.0 m vertically?
(a) Along the floor
𝑑
𝐹𝑃
𝑊𝑃 = 𝐹𝑃 𝑑 cos 𝜃
𝑊 = (230)(4.0)(1)
𝑊𝑃 = 𝑚𝑔𝑑 cos 0°
𝑊 = 920 J
𝑊𝑃 = (2200)(4.0)(1)
𝑚𝑔
𝑑
𝑚𝑔
𝑊𝑃 = 8800 J
7. At the top of a pole vault, an athlete pushes
the pole with a force given by ( ) = 150 −
190 2 acting over a distance of 0.20 m. How
much work is done on the athlete?
0.2
𝑊=
𝐹𝑃
𝑊𝑃 = 𝐹𝑃 𝑑 cos 𝜃
𝑁
𝑓
(b) Vertically
(150𝑥 − 190𝑥
𝐹𝑑𝑥 =
2 )𝑑𝑥
0
𝑊 = 75𝑥 −
2
𝑊 = 75(0.2)2 −
190
3
8. Calculate the amount of work done by a
force = 2 √ which acts on an object during
its journey along the axis from = 0.0 to
= 1.0 m.
𝑊=
2
√𝑥
1.0
𝑑𝑥 = 2
0
0.2
3
𝑥
0
190
(0.2)3 − 0 − 0
3
𝑊 = 2.4933 ≈ 2.5 𝐽
𝑊=
2𝑥1
2
(1 2)
1.0
0
𝑊 = 4(1 − 0)
𝑊 = 4.0 J
1
−
𝑥 2
𝑑𝑥
Name: _______________________
Mr. Kepple
Work-Energy Theorem
Work and Energy – HW#2
Date: ___________ Period: _____
1. The figure shows three forces applied to a trunk that moves leftward by
3.00 m over a frictionless floor. The force magnitudes are
N,
N,
N, and the indicated angle is
°. During the
displacement what is the change in kinetic energy of the trunk?
𝑊
𝐹 𝑑 cos 𝜃
( )( ) cos °
𝑊
𝐹 𝑑 cos 𝜃
( )( ) cos 2 °
−
𝑊
𝐹 𝑑 cos 𝜃
( )( ) cos
J
∆𝐾
J
°
∆𝐾
J
𝑊net
𝑊 +𝑊 +𝑊
∆𝐾
−
J
2. (a) How much work is done by the horizontal force
N
on the 18-kg block in the picture when the force pushes the block
5.0 m up along the 32° frictionless incline? (b) How much work is
done by the gravitational force on the block during this
displacement? (c) How much work is done by the normal force? (d)
What is the speed of the block (assume it was zero initially) after
this displacement?
(a) Work done by 𝐹𝑃
(b) Work by gravity
𝑊𝑃
𝑊𝐺
𝑊𝑃
𝐹𝑃 𝑑 cos 𝜃
(
𝑊𝑃
𝐹𝐺 𝑑 cos 𝜃
2
)( ) cos 2°
𝑊𝐺
𝑚𝑔𝑑 cos 𝜃
J≈ 4 J
𝑊𝐺
( 8)( 8)( ) cos 22° 𝑣
𝑊𝐺
−4 7 4 J ≈ −47 J
(c) Work by normal force
𝑊𝑁
(d) Speed of the block
𝐹𝑁 𝑑 cos
𝑚(𝑣 − 𝑣0 )
2𝑊
𝑚
𝑣
2(
𝑊
− 4 7 4)
( 8)
4 82 m/s ≈ 4 m/s
°
3. At an accident scene on a level road, investigators measure a car’s skid mark to be 98 m long. It was
a rainy day and the coefficient of friction was estimated to be 0.38. Use these data to determine the
speed of the car when the driver slammed on (and locked) the brakes.
𝑁
𝑑
−𝜇𝑚𝑔𝑑
𝑓
𝑚𝑔
𝑊
𝑓𝑑 cos 𝜃
𝜇𝑚𝑔𝑑 cos 8 °
𝑣
∆𝐾
𝑣
𝑚(𝑣 − 𝑣0 )
𝑚( − 𝑣0 )
𝑣
− 𝑚𝑣0
2𝜇𝑔𝑑
2( 8)( 8)( 8)
27 2 m/s ≈ 27 m/s
4. A 46.0-kg crate, starting from rest, is pulled across a floor with a constant horizontal force of 225 N.
For the first 11.0 m the floor is frictionless, and for the next 10.0 m the coefficient of friction is 0.20.
What is the final speed of the crate after being pulled these 21.0 m?
∆𝐾
𝑊
𝑚(𝑣 − 𝑣0 )
𝑚(𝑣 − )
𝑣
𝑊 +𝑊
𝐹𝑑 + 𝐹𝑑 − 𝜇𝑚𝑔𝑑
2 𝐹(𝑑 + 𝑑 ) − 𝜇𝑚𝑔𝑑
𝑚
𝑊
𝐹𝑑 cos °
𝐹𝑑
𝑊
𝐹𝑑 cos ° + 𝑓𝑑 cos 8 °
𝑊
𝐹𝑑 − 𝜇𝑚𝑔𝑑
2 (22 )(2 ) − ( 2)(4 )( 8)(
(4 )
𝑣
2 8 m/s ≈
)
m/s
5. A mass is attached to a spring which is held stretched a
distance by a force , and then released. The spring
compresses, pulling the mass. Determine the speed of the
mass when the spring returns: (a) to its normal length
(
); (b) to half its original extension ( /2). [Ignore friction]
(a) Speed at 𝑥
∆𝐾
2
(b) Speed at 𝑥
𝑊𝑆
𝑚(𝑣 − 𝑣0 )
𝑚𝑣 +
∆𝐾
2
𝑘𝑥
2
𝑚(𝑣 − 𝑣0 )
(𝑘𝑥)𝑑𝑥
𝑥/
𝐹
𝑥
𝑥
𝐹𝑥
𝑚
𝑊𝑆
𝑥
2
𝑣
𝑥/2
𝑚𝑣
𝑥
𝑘
2
𝑚𝑣
𝑘 (𝑥) −
𝐹
𝑥
𝑥
𝑚𝑣
𝑣
𝑥
𝑥/
𝑥
2
4
−
4 4
𝐹𝑥
4𝑚
Name: _______________________
Mr. Kepple
Chapter 7: Questions
Work and Energy – HW#3
Date: ___________ Period: _____
1. The figure shows two horizontal forces
that act on a block that is sliding to the right
across a frictionless floor. The graph shows
three plots of the block’s kinetic energy
versus time . Which of the plots best
corresponds to the following situations?
Justify each response.
(a)
(b)
(c)
Graph 2. The forces on
the block are balanced,
which means no
acceleration occurs. As
a result, the kinetic
energy is constant.
Graph 3. The net force
points opposite to the
direction of motion. The
object will eventually
come to rest and have
zero kinetic energy.
Graph 1. The net force
points parallel to the
direction of motion. The
object will gain speed
and the kinetic energy
will increase.
2. The figure shows four graphs (draw to the same
scale) of the component of a variable force
(directed along an axis) versus the position of a
particle on which the force acts. Rank the graphs
according to the work done by the force on the
particle from
to
, from most positive
work first to most negative work last.
Rank: b, a, c, d. The work done is the area
under the curve. Graph (b) is the only graph
with a net positive area. Graph (a) is next
since it has a net area of zero. Finally graph
(c) and then (d) since graph (c) has less
negative area than graph (d).
3. Spring is stiffer than spring
springs are compressed…
. The spring force of which spring does more work if the
(a) the same distance? Justify.
(b) by the same applied force? Justify.
Spring A. In order to compress each
spring by an equal length, more force
must be applied to the spring with the
greater stiffness constant. As a result,
the stiffer spring (spring A) does more
work.
Spring B. When both springs are
compressed by an equal applied force,
the spring with the smaller stiffness
constant will be compressed a greater
distance and as a result more work will
be done.
4. In the figure, a greased pig has a choice of three
frictionless slides along which to slide to the
ground. Rank the slides according to how much
work the gravitational force does on the pig during
the descent, greatest first. Justify your ranking.
All tie. Since gravity is a conservative force,
the work done by gravity will depend only on
the initial and final height of the pig. Along each of the three paths shown the pig will
start and end at the same height. Therefore the work done by gravity is the same along
all three paths.
5. In three situations, a briefly applied
horizontal force changes the velocity
of a hockey puck that slides over
frictionless ice. The overhead views of
the figure indicate, for each situation,
the puck’s initial speed , its final
speed , and the directions of the
corresponding velocity vectors. Rank
the situations according to the work done on the puck by the applied force, most positive first and
most negative last. Justify your ranking.
Ranking: c, b, a. According to the work-energy theorem, the work done on the object is
equal to the change in its kinetic energy. Situation (c) is the only situation with positive
work since the object gains speed. Situations (a) and (b) both have negative work since
the puck loses speed. Kinetic energy is proportional to speed squared so more
negative work is done in (a) since the magnitude of the speeds are greater than in (b).
6. The figure gives the component
particle. If the particle begins at rest
when it has… (justify each response)
of a force that can act on a
, what is its coordinate
(a) its greatest kinetic energy?
3 m. The gain in kinetic energy will be equal to the work
done, in other words the area under the curve.
(b) its greatest speed?
3 m. Kinetic energy depends on speed. The location with
the most kinetic energy also has the most speed.
(c) zero speed?
(d) What is the particle’s direction of travel
after it reaches
m?
m and m. The object starts at rest
and returns to rest when the net work
done is equal to zero. The area under
the curve from 0 m to 6 m is zero.
Negative 𝑥. The object returns to rest at
6 m and past this point is acted on by a
negative net force. It will accelerate and
gain speed in the negative 𝑥 direction.
Potential Energy
Mr. Kepple
Name: _______________________
Work and Energy – HW#4
Date: ___________ Period: _____
1. A particular spring obeys the force law ⃑
.
(a) Is this force conservative? Explain.
(b) Determine the form of the potential energy
function.
Yes this force is conservative. Since the
force is a function of position, the work
done by the force depends only on the
initial and final position (the end points) of
the object. Therefore, by definition, the
force is conservative.
𝑈
𝐹 𝑑𝑥
𝑈
( 𝑘𝑥
𝑈
1
𝑘𝑥
1
𝑎𝑥
𝑏𝑥 ) 𝑑𝑥
𝑎𝑥
1
5
𝑏𝑥 5
𝐶
2. The potential energy of an object is given by ( )
(a) What is the force, ⃑ ?
(b) What force acts on the object at
𝐹(𝑥)
𝑑𝑈(𝑥)
𝑑𝑥
𝐹(𝑥)
6𝑥
𝐹( )
6( )
𝐹(𝑥)
𝑑
( 𝑥
𝑑𝑥
𝐹( )
8N
𝐹(𝑥)
(6𝑥
𝑥
m?
)
)
3. A particle constrained to move in one dimension is acted on by the force ( )
where is a
constant with units appropriate to the SI system. Find the potential energy function ( ), if is
arbitrarily defined to be zero at
m, so that ( )
.
𝑈
𝐹 𝑑𝑥
𝑈
𝑘
𝑥
𝑑𝑥
𝑥 − 𝑑𝑥
𝑈( )
𝑈
𝑘
𝑥
𝐶
𝑘
( )
𝑈
𝑘
𝑥
𝑘
8
𝐶
𝑈
𝑘
𝐶
𝑘
8
J
4. The figure shows a plot of potential energy versus
position of a 2.0-kg particle that can travel only along an
axis. Assume nonconservative forces are not involved.
Suppose the particle is released at
speed of 7.0 m/s, headed in the negative
J
with an initial
direction.
J
Remember the work-energy theorem! Since
and
this means that
. In other words, when
an object gains potential energy it loses kinetic energy.
(a) Does the particle have enough kinetic
energy to reach
m?
𝑈( )
𝑈
𝑈( )
𝑈
J
1
𝐾
(b) What is the magnitude and direction of the
force acting on the particle as it begins to move
to the left of
m?
𝑚𝑣
1
𝐹
( )(7)
𝑑𝑈
𝑑𝑥
9J
𝐹
Yes the particle can reach 𝑥
. It
needs at least 30 J of kinetic energy
and it has 49 J.
5. A spring has a spring constant
35.0 J of potential energy?
𝑈𝑆
1
𝐹
N to the right
of 82.0 N/m. How much must this spring be compressed to store
𝑘𝑥
𝑈𝑆
𝑘
𝑥
𝑈
𝑥
( 5)
8
9
𝑥≈
9 6
9
6. A 1.60-m tall person lifts a 1.95-kg book off the ground so it is 2.20 m above the ground. What is the
potential energy of the book relative to (a) the ground, and (b) the top of the person’s head?
6
book
(a) relative to the ground
(b) relative to the head
head
𝑈
𝑚𝑔ℎ
𝑈
𝑚𝑔ℎ
𝑈
( 95)(9 8)(
𝑈
( 95)(9 8)(
𝑈
66 J
ground
𝑈
𝑈≈
J
J
)
𝑈≈
5J
6)