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  • Apply problem-solving techniques to solve for quantities in more complex systems of forces.
  • Integrate concepts from kinematics to solve problems using Newton's laws of motion.

There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills.

Drag force on a barge

Suppose two tugboats push on a barge at different angles, as shown in [link] . The first tugboat exerts a force of 2.7 × 10 5 N size 12{2 "." 7 times "10" rSup { size 8{5} } " N"} {} in the x -direction, and the second tugboat exerts a force of 3.6 × 10 5 N size 12{3 "." 6 times "10" rSup { size 8{5} } " N"} {} in the y -direction.

(a) A view from above two tugboats pushing on a barge. One tugboat is pushing with the force F sub x equal to two point seven multiplied by ten to the power five newtons, shown by a vector arrow acting toward the right in the x direction. Another tugboat is pushing with a force F sub y equal to three point six multiplied by ten to the power five newtons acting upward in the positive y direction. Acceleration of the barge, a, is shown by a vector arrow directed fifty-three point one degree angle above the x axis. In the free-body diagram, F sub y is acting on a point upward, F sub x is acting toward the right, and F sub D is acting approximately southwest. (b) A right triangle is made by the vectors F sub x and F sub y. The base vector is shown by the force vector F sub x. and the perpendicular vector is shown by the force vector F sub y. The resultant is the hypotenuse of this triangle, making a fifty-three point one degree angle from the base, shown by the vector force F sub net pointing up the inclination. A vector F sub D points down the incline.
(a) A view from above of two tugboats pushing on a barge. (b) The free-body diagram for the ship contains only forces acting in the plane of the water. It omits the two vertical forces—the weight of the barge and the buoyant force of the water supporting it cancel and are not shown. Since the applied forces are perpendicular, the x - and y -axes are in the same direction as F x size 12{F rSub { size 8{x} } } {} and F y size 12{F rSub { size 8{y} } } {} . The problem quickly becomes a one-dimensional problem along the direction of F app size 12{F rSub { size 8{"app"} } } {} , since friction is in the direction opposite to F app size 12{F rSub { size 8{"app"} } } {} .

If the mass of the barge is 5.0 × 10 6 kg size 12{5 times "10" rSup { size 8{6} } " kg"} {} and its acceleration is observed to be 7 . 5 × 10 2 m/s 2 size 12{7 "." "52" times "10" rSup { size 8{ - 2} } " m/s" rSup { size 8{2} } } {} in the direction shown, what is the drag force of the water on the barge resisting the motion? (Note: drag force is a frictional force exerted by fluids, such as air or water. The drag force opposes the motion of the object.)

Strategy

The directions and magnitudes of acceleration and the applied forces are given in [link] (a) . We will define the total force of the tugboats on the barge as F app size 12{F rSub { size 8{"app"} } } {} so that:

F app = F x + F y size 12{F rSub { size 8{ ital "app"} } ital "= F" rSub { size 8{x} } ital "+ F" rSub { size 8{y} } } {}

Since the barge is flat bottomed, the drag of the water F D size 12{F rSub { size 8{D} } } {} will be in the direction opposite to F app size 12{F rSub { size 8{"app"} } } {} , as shown in the free-body diagram in [link] (b). The system of interest here is the barge, since the forces on it are given as well as its acceleration. Our strategy is to find the magnitude and direction of the net applied force F app size 12{F rSub { size 8{"app"} } } {} , and then apply Newton’s second law to solve for the drag force F D size 12{F rSub { size 8{D} } } {} .

Solution

Since F x size 12{F rSub { size 8{x} } } {} and F y size 12{F rSub { size 8{y} } } {} are perpendicular, the magnitude and direction of F app size 12{F rSub { size 8{"app"} } } {} are easily found. First, the resultant magnitude is given by the Pythagorean theorem:

F app = F x 2 + F y 2 F app = ( 2.7 × 10 5 N ) 2 + ( 3.6 × 10 5 N ) 2 = 4.5 × 10 5 N. alignl { stack { size 12{F rSub { size 8{ ital "app"} } = \( F rSub { size 8{x} rSup { size 8{2} } } + F rSub { size 8{y} rSup { size 8{2} } } \) rSup { size 8{1/2} } } {} #F rSub { size 8{ ital "app"} } = \( \( 2 "." 7 times "10" rSup { size 8{5} } " N" \) rSup { size 8{2} } + \( 3 "." 6 times "10" rSup { size 8{5} } " N" \) rSup { size 8{2} } \) rSup { size 8{1/2} } =4 "." "50" times "10" rSup { size 8{5} } " N" "." {} } } {}

The angle is given by

θ = tan 1 F y F x θ = tan 1 3.6 × 10 5 N 2.7 × 10 5 N = 53º , alignl { stack { size 12{θ="tan" rSup { size 8{ - 1} } left ( { {F rSub { size 8{y} } } over {F rSub { size 8{x} } } } right )} {} #θ="tan" rSup { size 8{ - 1} } left ( { { \( 2 "." 7 times "10" rSup { size 8{5} } " N" \) } over { \( 3 "." 6 times "10" rSup { size 8{5} } " N" \) } } right )="53" "." 1°, {} } } {}

which we know, because of Newton’s first law, is the same direction as the acceleration. F D size 12{F rSub { size 8{D} } } {} is in the opposite direction of F app size 12{F rSub { size 8{"app"} } } {} , since it acts to slow down the acceleration. Therefore, the net external force is in the same direction as F app size 12{F rSub { size 8{"app"} } } {} , but its magnitude is slightly less than F app size 12{F rSub { size 8{"app"} } } {} . The problem is now one-dimensional. From [link] (b) , we can see that

F net = F app F D size 12{F rSub { size 8{"net"} } =F rSub { size 8{"app"} } - F rSub { size 8{D} } } {} .

But Newton’s second law states that

F net = ma size 12{F rSub { size 8{"net"} } = ital "ma"} {} .

Thus,

F app F D = ma size 12{F rSub { size 8{"app"} } - F rSub { size 8{D} } = ital "ma"} {} .

This can be solved for the magnitude of the drag force of the water F D size 12{F rSub { size 8{D} } } {} in terms of known quantities:

F D = F app ma size 12{F rSub { size 8{D} } =F rSub { size 8{"app"} } - ital "ma"} {} .

Substituting known values gives

F D = ( 4 . 5 × 10 5 N ) ( 5 . 0 × 10 6 kg ) ( 7 . 5 × 10 –2 m/s 2 ) = 7 . 5 × 10 4 N size 12{F rSub { size 8{D} } = \( 4 "." "50" times "10" rSup { size 8{5} } " N" \) - \( 5 "." "00" times "10" rSup { size 8{6} } " kg" \) \( 7 "." "50" times "10" rSup { size 8{"-2"} } " m/s" rSup { size 8{2} } \) =7 "." "50" times "10" rSup { size 8{4} } " N"} {} .

The direction of F D size 12{F rSub { size 8{D} } } {} has already been determined to be in the direction opposite to F app size 12{F rSub { size 8{"app"} } } {} , or at an angle of 53º size 12{"53" "." 1°} {} south of west.

Questions & Answers

how does sound affect temperature
Clement Reply
how to solve wave question
Wisdom Reply
I would like to know how I am not at all smart when it comes to math. please explain so I can understand. sincerly
Emma
Just know d relationship btw 1)wave length 2)frequency and velocity
Talhatu
First of all, you are smart and you will get it👍🏽... v = f × wavelength see my youtube channel: "mathwithmrv" if you want to know how to rearrange equations using the balance method
PhysicswithMrV
nice self promotion though xD
Beatrax
thanks dear
Chuks
hi pls help me with this question A ball is projected vertically upwards from the top of a tower 60m high with a velocity of 30ms1.what is the maximum height above the ground level?how long does it take to reach the ground level?
mahmoud
please guys help, what is the difference between concave lens and convex lens
Vincent Reply
convex lens brings rays of light to a focus while concave diverges rays of light
Christian
for mmHg to kPa yes
Matthew
it depends on the size
Matthew Reply
please what is concave lens
Vincent
a lens which diverge the ray of light
rinzuala
concave diverges light
Matthew
thank you guys
Vincent
A diverging lens
Yusuf
What is isotope
Yusuf
each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
Karthi
what is wire wound resistors?
Naveedkhan Reply
What are the best colleges to go to for physics
Matthew Reply
I would like to know this too
Trevor
How do I calculate uncertainty in a frequency?
Rebecca Reply
Calculate . ..
Olufunsho
What is light wave
Sakeenah Reply
What is wave
Sakeenah
What is light
Sakeenah
okay
True
explain how neurons communicate feed and stimulate
Jeff
Great science students
Omo
A wave is a disturbance which travels through the medium transferring energy from one form to another without causing any permanent displacement of d medium itself
OGOR
Light is a form o wave
OGOR
Neurons communicate by sending message through nerves in coordination
OGOR
What are petrochemicals, give two examples
OGOR
light has dual nature, particle as well as wave. when we want to explain phenomena like Interference of light, then we consider light as wave.
Lalita
what is it as in the form of it or how to visualize it or what it contains
Matthew
particles of light are like small packets of energy called photons, and flow or motion of photons is wave like
Lalita
light is just the energy of which photons emit
Matthew
the wave is how they travel
Matthew
photons do not emitt energy, they are energy. They are massless particles.
Lalita
a wave is a disturbance through the medium. Have you ever thrown a stone in still water? the disturbance produced travels in form of wave, the wave produced by throwing stone in still water are circular in nature.
Lalita
a photon does contain mass when in motion. it doesnt contain mass when at rest
Matthew
when would it ever be at rest
Bob
a wave is a disturbance of which energy travels
Matthew
that's darkness. darkness has no mass because the photons within in aren't moving or producing energy
Matthew
Hi guys. Please I've been trying to understand the concept of SHM, but it's not been really easy, could someone please explain it to me or suggest a site I could visit? Thank you.
Odo
***google.com/url?sa=t&source=web&rct=j&url=***hyperphysics.phy-astr.gsu.edu/hbase/shm.html&ved=2ahUKEwiwu_PYqfzdAhXOmVkKHXi2CvkQFjAPegQIARAB&usg=AOvVaw0h5IZmrkQK4KtEMygT4ZGK
Matthew
effective mass of photons only comes into picture when we consider it accelerating in gravitational field, mass of photon has no meaning as it is always travelling with speed of light and is never at rest. with that high speed, Energy and momentum are equivalent. and darkness is absense of photons.
Lalita
darkness is absense of light. not the presence of 'resting photons'. photons are never at rest.
Lalita
photons are present in darkness but don't give off any light because they are stationary with no mass or energy. once a force makes them move again they will gain mass and give off light
Matthew
this theory is presented in Einsteins theory of special relativity
Matthew
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density and the viscosity iter of the liquid .use the dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square ×p all over 4×iter ×l
True
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density (rho)and the viscosity (iter)of the liquid. Use the method of dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square x p all over 4 x iter x l
True
Matthew, photons ARE light. there is no such thing as a photon that isn't moving. in fact the speed they move at is called C (for constant) in physics. through a vacuum they always travel at this speed no matter what. they can not slow down; except in another medium.
Brad
The reason why a photon can go at this speed is BECAUSE it had no mass. nothing can go this speed or faster because it needs to have no mass or negative mass. that's why it's called the constant.
Brad
when a photon hits something that is opaque, this is the only way to "stop"it. it isn't merely stopped but absorbed and turned into heat energy, then the remaining energy is reflected in different wavelengths. that reflection is what we call color. the darker something is, the less photons are ther
Brad
e. complete blackness is the absolute absence of photons altogether. I believe what you're referring to is not speed, but wavelength, which is indirectly proportional to the amount of energy a particular photon is made up of.
Brad
in order for a photon to have zero wavelength, it must (at least theoretically) have infinite energy.
Brad
about mass: you may have photons confused with electrons. elections have a mass so small that people say they are without mass, but they do. it is called electron mass or Me-.
Brad
you may also be getting electrons and photons confused because of the cherenkov effect. that is what happens when a particle travels faster than light IN THAT PARTICULAR MEDIUM. I emphasize that because no other particle besides photons can go the speed of c.
Brad
when a particle goes faster than light in a particular medium, a blue light is emitted, called cherenkov radiation. this is why nuclear reactors glow blue.
Brad
nuclear reactors release so much energy that when they emit electrons, those electrons are given enough energy to go faster than light in that medium (in this case water), releasing blue light. if you put the reactor in air or a vacuum, this effect wouldn't happen because the speed of light in air
Brad
is very close to c, which is the universal speed limit. I'd you did go faster than c, time would go backwards and you would have infinite theoretical mass and probably spagghettify, like with a black hole.
Brad
*if
Brad
*electrons
Brad
light waves can travel through a vacuum, and do not require a medium. In empty space, the wave does not dissipate (grow smaller) no matter how far it travels, because the wave is not interacting with anything else.
Salim
Please is there any instructional material for sounds Waves, Echo, light waves
Salami
how far there is hot topic that is boarding me now
Abraham
linear motion
Ahmed
kinematic
Abraham
tell us about it
Akinsanya
kinematic
Emma
kinematics disscuss the motion without cuases ...
ghulam
wow I like what am seeing here I need someone to brush me up in physics in fact I'll say I know nothing
Godslight
How does the Geiger tube works
Salma Reply
pls he do we find for tension
Belinda Reply
tension is equal to the weight of the object. so for example if something weighs 45 Newtons then the tension in the Rope holding it is 45 Newtons. and because it is in equilibrium if the object is 45N and there are three ropes holding it there would be 15 N of tension in each to equal the weight
Shii
does that work for you?
Shii
tnx
Belinda
very correct
Kudzy
A prankster applies 450 V to an 80.0 µF capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?
Khanh Reply
what is mass
Aliyu Reply
the quantity of matter that a body contains, as measured by its acceleration under a given force or by the force exerted on it by a gravitational field.
Aliyu
I agree wth aliyu shuwa
Nikita
correct
Jalil
or in easier terms the amount of stuff in an object (stuff meaning whatever element or material that makes the object heavy) the object composition
Shii
an object's resistance to change in motion?
Kudzy
your mass never changes but your weight changes based on the gravitational pull of a system or planet. your mass is just the amount of matter with a certain object
Matthew
please can someone help, why a bats can fly in the night without heating anything, how does he managed an see in darkness?
Vincent
...hit an obstacle and has bounced back.
Odo
Hi Vincent. From what I can remember and what I've learnt, bats do not have a good eye sight and so they make use of waves, when they send out waves and they do not return back to them, they realise that the site is free of obstacles but if it sends a wave and it returns back, it means it hit an
Odo
obstacle and bounced back
Odo
So bats make use of waves in place of eye sight. I think but confirm from a few sites .
Odo
they us echolocation to make a sort of mental map. many sea animals use the same method. this can also be compared to sonar which works in the same way
Matthew
y is atom d smallest particle
EDWIN Reply
yes
Lajpat
its a question Lajpat Rai
EDWIN
Google Quantum Physics or refer to the text
Shii
atoms are considered the smallest unit of matter. it is further divided into three subatomic particles: protons, neutrons, and electrons.
Arshiya
what's the dimensions of moment of inertia?
Rose Reply
what is the same conditions
Naveedkhan Reply
what is important property of thermistor
Naveedkhan

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