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Solving problems is an essential part of the understanding process.

Questions and their answers are presented here in the module text format as if it were an extension of the treatment of the topic. The idea is to provide a verbose explanation, detailing the application of theory. Solution presented is, therefore, treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.

Representative problems and their solutions

We discuss problems, which highlight certain aspects of the study leading to the uniform circular motion. The questions are categorized in terms of the characterizing features of the subject matter :

  • Direction of velocity
  • Direction of position vector
  • Velocity
  • Relative speed
  • Nature of UCM

Direction of velocity

Problem : A particle moves in xy-plane along a circle of radius "r". The particle moves at a constant speed in anti-clockwise direction with center of circle as the origin of the coordinate system. At a certain instant, the velocity of the particle is i – √3 j . Determine the angle that velocity makes with x-direction.

Solution : The sign of y-component of velocity is negative, whereas that of x-component of velocity is positive. It means that the particle is in the third quadrant of the circle as shown in the figure.

Top view of uniform circular motion in xy-plane

The acute angle formed by the velocity with x-axis is obtained by considering the magnitude of components (without sign) as :

tan α = v y v x = 3 1 = 3 = tan 60 0

α = 60 0

This is the required angle as measured in clockwise direction from x-axis. If the angle is measured in anti-clockwise direction from positive direction of x-axis, then

α = 360 0 60 0 = 300 0

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Direction of position vector

Problem : A particle moves in xy-plane along a circle of radius “r”. The particle moves at a constant speed in anti-clockwise direction with center of circle as the origin of the coordinate system. At a certain instant, the velocity of the particle is i – √3 j . Determine the angle that position vector makes with x-direction.

Solution : The sign of y-component of velocity is negative, whereas that of x-component of velocity is positive. It means that the particle is in the third quadrant of the circle as shown in the figure.

Top view of uniform circular motion in xy-plane

The acute angle formed by the velocity with x-axis is obtained by considering the magnitude of components (without sign) as :

tan α = v y v x = 3 1 = 3 = tan 60 0

α = 60 0

But, we know that position vector is perpendicular to velocity vector. By geometry,

θ = 180 0 30 0 = 150 0

This is the angle as measured in clockwise direction from x-axis. If the angle is measured in anti-clockwise direction from positive direction of x-axis, then

α = 360 0 150 0 = 210 0

Note : Recall the derivation of the expression of velocity vector in the previous module. We had denoted “θ” as the angle that position vector makes with x-axis (not the velocity vector). See the figure that we had used to derive the velocity expression.

Top view of uniform circular motion in xy-plane

As a matter of fact “θ” is the angle that velocity vector makes with y-axis (not x-axis). We can determine the angle “θ” by considering the sign while evaluating tan θ,

tan θ = v x v y = 1 3 = tan 150 0

θ = 150 0

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Velocity

Problem : A particle moves with a speed 10 m/s in xy-plane along a circle of radius 10 m in anti-clockwise direction. The particle starts moving with constant speed from position (r,0), where "r" denotes the radius of the circle. Find the velocity of the particle (in m/s), when its position makes an angle 135° with x – axis.

Solution : The velocity of the particle making an angle "θ" with x – axis is given as :

Uniform circular motion

v = v x i + v y j = v sin θ i + v cos θ j

Here,

v x = - v sin θ = - 10 sin 135 0 = - 10 x ( 1 2 ) = - 5 2 v y = v cos θ = 10 cos 135 0 = 10 x ( - 1 2 ) = - 5 2

Here, both the components are negative.

v = v x i + v y j v = - ( 5 2 i + 5 2 j ) m / s

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Relative speed

Problem : Two particles tracing a circle of radius 10 m begin their journey simultaneously from a point on the circle in opposite directions. If their speeds are 2.0 m/s and 1.14 m/s respectively, then find the time after which they collide.

Solution : The particles approach each other with a relative speed, which is equal to the sum of their speeds.

v r e l = 2.0 + 1.14 = 3.14 m / s

For collision to take place, the particles need to cover the initial separation with the relative speed as measured above. The time for collision is, thus, obtained as :

t = 2 π r v r e l = 2 x 3.14 x 10 3.14 = 20 s

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Nature of ucm

Problem : Two particles “A” and “B” are moving along circles of radii " r A " and " r B " respectively at constant speeds. If the particles complete one revolution in same time, then prove that speed of the particle is directly proportional to radius of the circular path.

Solution : As the time period of the UCM is same,

T = 2 π r A v A = 2 π r B v B

v A r A = v B r B

v A v B = r A r B

Hence, speed of the particle is directly proportional to the radius of the circle.

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Problem : Two particles “A” and “B” are moving along circles of radii " r A " and " r B " respectively at constant speeds. If the particles have same acceleration, then prove that speed of the particle is directly proportional to square root of the radius of the circular path.

Solution : As the acceleration of the UCM is same,

v A 2 r A = v B 2 r B

v A 2 v B 2 = r A r B

v A v B = r A r B

Hence, speed of the particle is directly proportional to square root of the radius of the circular path.

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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?
s. Reply
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Kinematics fundamentals. OpenStax CNX. Sep 28, 2008 Download for free at http://cnx.org/content/col10348/1.29
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