# 3.7 Further applications of newton’s laws of motion  (Page 2/6)

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Discussion

The numbers used in this example are reasonable for a moderately large barge. It is certainly difficult to obtain larger accelerations with tugboats, and small speeds are desirable to avoid running the barge into the docks. Drag is relatively small for a well-designed hull at low speeds, consistent with the answer to this example, where ${F}_{\text{D}}$ is less than 1/600th of the weight of the ship.

In the earlier example of a tightrope walker we noted that the tensions in wires supporting a mass were equal only because the angles on either side were equal. Consider the following example, where the angles are not equal; slightly more trigonometry is involved.

## Different tensions at different angles

Consider the traffic light (mass 15.0 kg) suspended from two wires as shown in [link] . Find the tension in each wire, neglecting the masses of the wires. A traffic light is suspended from two wires. (b) Some of the forces involved. (c) Only forces acting on the system are shown here. The free-body diagram for the traffic light is also shown. (d) The forces projected onto vertical ( y ) and horizontal ( x ) axes. The horizontal components of the tensions must cancel, and the sum of the vertical components of the tensions must equal the weight of the traffic light. (e) The free-body diagram shows the vertical and horizontal forces acting on the traffic light.

Strategy

The system of interest is the traffic light, and its free-body diagram is shown in [link] (c). The three forces involved are not parallel, and so they must be projected onto a coordinate system. The most convenient coordinate system has one axis vertical and one horizontal, and the vector projections on it are shown in part (d) of the figure. There are two unknowns in this problem ( ${T}_{1}$ and ${T}_{2}$ ), so two equations are needed to find them. These two equations come from applying Newton’s second law along the vertical and horizontal axes, noting that the net external force is zero along each axis because acceleration is zero.

Solution

First consider the horizontal or x -axis:

${F}_{\text{net}x}={T}_{\text{2}x}-{T}_{\text{1}x}=0.$

Thus, as you might expect,

${T}_{\text{1}x}={T}_{\text{2}x}.$

This gives us the following relationship between ${T}_{1}$ and ${T}_{2}$ :

${T}_{1}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left(\text{30º}\right)={T}_{2}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\left(\text{45º}\right).$

Thus,

${T}_{2}=\left(1\text{.}\text{225}\right){T}_{1}.$

Note that ${T}_{1}$ and ${T}_{2}$ are not equal in this case, because the angles on either side are not equal. It is reasonable that ${T}_{2}$ ends up being greater than ${T}_{1}$ , because it is exerted more vertically than ${T}_{1}$ .

Now consider the force components along the vertical or y -axis:

${F}_{\text{net}\phantom{\rule{0.25em}{0ex}}y}={T}_{\text{1}y}+{T}_{\text{2}y}-w=0.$

This implies

${T}_{\text{1}y}+{T}_{\text{2}y}=w.$

Substituting the expressions for the vertical components gives

${T}_{1}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\left(\text{30º}\right)+{T}_{2}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\left(\text{45º}\right)=w.$

There are two unknowns in this equation, but substituting the expression for ${T}_{2}$ in terms of ${T}_{1}$ reduces this to one equation with one unknown:

${T}_{1}\left(0\text{.}\text{500}\right)+\left(1\text{.}\text{225}{T}_{1}\right)\left(0\text{.}\text{707}\right)=w=\text{mg},$

which yields

$\left(1\text{.}\text{366}\right){T}_{1}=\left(\text{15}\text{.}\text{0 kg}\right)\left(9\text{.}{\text{80 m/s}}^{2}\right).$

Solving this last equation gives the magnitude of ${T}_{1}$ to be

${T}_{1}=\text{108 N}.$

Finally, the magnitude of ${T}_{2}$ is determined using the relationship between them, ${T}_{2}$ = 1.225 ${T}_{1}$ , found above. Thus we obtain

${T}_{2}=\text{132 N}.$

Discussion

Both tensions would be larger if both wires were more horizontal, and they will be equal if and only if the angles on either side are the same (as they were in the earlier example of a tightrope walker).

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
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Renato
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?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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