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a ( t ) = d v ( t ) d t = A ω 2 cos ω t i ^ A ω 2 sin ω t j ^ .

From this equation we see that the acceleration vector has magnitude A ω 2 and is directed opposite the position vector, toward the origin, because a ( t ) = ω 2 r ( t ) .

Circular motion of a proton

A proton has speed 5 × 10 6 m/s and is moving in a circle in the xy plane of radius r = 0.175 m. What is its position in the xy plane at time t = 2.0 × 10 −7 s = 200 ns? At t = 0, the position of the proton is 0.175 m i ^ and it circles counterclockwise. Sketch the trajectory.

Solution

From the given data, the proton has period and angular frequency:

T = 2 π r v = 2 π ( 0.175 m ) 5.0 × 10 6 m / s = 2.20 × 10 −7 s
ω = 2 π T = 2 π 2.20 × 10 −7 s = 2.856 × 10 7 rad / s .

The position of the particle at t = 2.0 × 10 −7 s with A = 0.175 m is

r ( 2.0 × 10 −7 s ) = A cos ω ( 2.0 × 10 −7 s ) i ^ + A sin ω ( 2.0 × 10 −7 s ) j ^ m = 0.175 cos [ ( 2.856 × 10 7 rad / s ) ( 2.0 × 10 −7 s ) ] i ^ + 0.175 sin [ ( 2.856 × 10 7 rad / s ) ( 2.0 × 10 −7 s ) ] j ^ m = 0.175 cos ( 5.712 rad ) i ^ + 0.175 sin ( 5.712 rad ) j ^ = 0.147 i ^ 0.095 j ^ m .

From this result we see that the proton is located slightly below the x -axis. This is shown in [link] .

A graph of y position as a function of x position is shown. Both x and y are measured in meters and run from -0.2 to 0.2. A proton is moving in a counterclockwise circle centered on the origin is shown at 11 different times. At t = 0 s the particle is at x = 0.175 m and y = 0. At t = 200 nanoseconds, the particle is at a position given by vector 0.147 I hat minus 0.95 j hat meters.
Position vector of the proton at t = 2.0 × 10 −7 s = 200 ns . The trajectory of the proton is shown. The angle through which the proton travels along the circle is 5.712 rad, which a little less than one complete revolution.

Significance

We picked the initial position of the particle to be on the x- axis. This was completely arbitrary. If a different starting position were given, we would have a different final position at t = 200 ns.

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Nonuniform circular motion

Circular motion does not have to be at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of the motion.

In uniform circular motion, the particle executing circular motion has a constant speed and the circle is at a fixed radius. If the speed of the particle is changing as well, then we introduce an additional acceleration in the direction tangential to the circle. Such accelerations occur at a point on a top that is changing its spin rate, or any accelerating rotor. In Displacement and Velocity Vectors we showed that centripetal acceleration is the time rate of change of the direction of the velocity vector. If the speed of the particle is changing, then it has a tangential acceleration    that is the time rate of change of the magnitude of the velocity:

a T = d | v | d t .

The direction of tangential acceleration is tangent to the circle whereas the direction of centripetal acceleration is radially inward toward the center of the circle. Thus, a particle in circular motion with a tangential acceleration has a total acceleration    that is the vector sum of the centripetal and tangential accelerations:

a = a C + a T .

The acceleration vectors are shown in [link] . Note that the two acceleration vectors a C and a T are perpendicular to each other, with a C in the radial direction and a T in the tangential direction. The total acceleration a points at an angle between a C and a T .

The acceleration of a particle on a circle is shown along with its radial and tangential components. The centripetal acceleration a sub c points radially toward the center of the circle. The tangential acceleration a sub T is tangential to the circle at the particle’s position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.
The centripetal acceleration points toward the center of the circle. The tangential acceleration is tangential to the circle at the particle’s position. The total acceleration is the vector sum of the tangential and centripetal accelerations, which are perpendicular.

Questions & Answers

who discover periodic table?
lovet Reply
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
Oniyide Reply
different types of wave
Yog Reply
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
Akshay Reply
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
Vijaybhai Reply
motion in straight line is kinematic's part
Ritik
yea
Manoj
what is Lorentz force?
jyotirmayee Reply
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay Reply
what answer fir this
Akshay
what's the working difference between a dynamo and a pump?
Piyali Reply
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
Piyali Reply
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
from where
Nirupam
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
jyotirmayee
Kota Rajasthan
Nirupam
so strange,,,,,r u preparing for pmt,,,?
jyotirmayee
for IIT
Nirupam
then concern near book store
jyotirmayee
in Rajasthan
jyotirmayee
or ask questions here
jyotirmayee
thanks for your help
Nirupam
its fine,,most welcome
jyotirmayee
what is three dimensional coordinate system?
Sachindra Reply
considering the change of vectors in all three dimensions of space
vedanth
direction co-sign of vector questions
Vipin Reply
opposite
Sonu
opposite mean
Vipin
what is the different between action and reaction?
Asali Reply
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
Yoblaze
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
Amalesh
or we can say , reaction is the result of action
Ritik
Practice Key Terms 4

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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