<< Chapter < Page Chapter >> Page >

A particle moves on a circular path of radius b according to the function r ( t ) = b cos ( ω t ) i + b sin ( ω t ) j , where ω is the angular velocity, d θ / d t .

This figure is the graph of a circle centered at the origin with radius of 3. The orientation of the circle is counter-clockwise. Also, in the fourth quadrant there are two vectors. The first starts on the circle and terminates at the origin. The second vector is tangent at the same point in the fourth quadrant towards the x-axis.

Find the velocity function and show that v ( t ) is always orthogonal to r ( t ) .

r ' ( t ) = b ω sin ( ω t ) i + b ω cos ( ω t ) j . To show orthogonality, note that r ' ( t ) · r ( t ) = 0 .

Got questions? Get instant answers now!

Show that the speed of the particle is proportional to the angular velocity.

Got questions? Get instant answers now!

Evaluate d d t [ u ( t ) × u ( t ) ] given u ( t ) = t 2 i 2 t j + k .

0 i + 2 j + 4 t j

Got questions? Get instant answers now!

Find the antiderivative of r ' ( t ) = cos ( 2 t ) i 2 sin t j + 1 1 + t 2 k that satisfies the initial condition r ( 0 ) = 3 i 2 j + k .

Got questions? Get instant answers now!

Evaluate 0 3 t i + t 2 j d t .

1 3 ( 10 3 / 2 1 )

Got questions? Get instant answers now!

An object starts from rest at point P ( 1 , 2 , 0 ) and moves with an acceleration of a ( t ) = j + 2 k , where a ( t ) is measured in feet per second per second. Find the location of the object after t = 2 sec.

Got questions? Get instant answers now!

Show that if the speed of a particle traveling along a curve represented by a vector-valued function is constant, then the velocity function is always perpendicular to the acceleration function.


v ( t ) = k v ( t ) · v ( t ) = k d d t ( v ( t ) · v ( t ) ) = d d t k = 0 v ( t ) · v ' ( t ) + v ' ( t ) · v ( t ) = 0 2 v ( t ) · v ' ( t ) = 0 v ( t ) · v ' ( t ) = 0 .
The last statement implies that the velocity and acceleration are perpendicular or orthogonal.

Got questions? Get instant answers now!

Given r ( t ) = t i + 3 t j + t 2 k and u ( t ) = 4 t i + t 2 j + t 3 k , find d d t ( r ( t ) × u ( t ) ) .

Got questions? Get instant answers now!

Given r ( t ) = t + cos t , t sin t , find the velocity and the speed at any time.

v ( t ) = 1 sin t , 1 cos t , speed = v ( t ) = 4 2 ( sin t + cos t )

Got questions? Get instant answers now!

Find the velocity vector for the function r ( t ) = e t , e t , 0 .

Got questions? Get instant answers now!

Find the equation of the tangent line to the curve r ( t ) = e t , e t , 0 at t = 0 .

x 1 = t , y 1 = t , z 0 = 0

Got questions? Get instant answers now!

Describe and sketch the curve represented by the vector-valued function r ( t ) = 6 t , 6 t t 2 .

Got questions? Get instant answers now!

Locate the highest point on the curve r ( t ) = 6 t , 6 t t 2 and give the value of the function at this point.

r ( t ) = 18 , 9 at t = 3

Got questions? Get instant answers now!

The position vector for a particle is r ( t ) = t i + t 2 j + t 3 k . The graph is shown here:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve begins at the bottom of the box to the left and curves upward to the top right corner.

Find the velocity vector at any time.

Got questions? Get instant answers now!

Find the speed of the particle at time t = 2 sec.

593

Got questions? Get instant answers now!

Find the acceleration at time t = 2 sec.

Got questions? Get instant answers now!

A particle travels along the path of a helix with the equation r ( t ) = cos ( t ) i + sin ( t ) j + t k . See the graph presented here:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve is a helix and begins at the bottom of the box to the right and spirals upward.

Find the following:

Velocity of the particle at any time

v ( t ) = sin t , cos t , 1

Got questions? Get instant answers now!

Speed of the particle at any time

Got questions? Get instant answers now!

Acceleration of the particle at any time

a ( t ) = cos t i sin t j + 0 j

Got questions? Get instant answers now!

Find the unit tangent vector for the helix.

Got questions? Get instant answers now!

A particle travels along the path of an ellipse with the equation r ( t ) = cos t i + 2 sin t j + 0 k . Find the following:

Velocity of the particle

v ( t ) = sin t , 2 cos t , 0

Got questions? Get instant answers now!

Speed of the particle at t = π 4

Got questions? Get instant answers now!

Acceleration of the particle at t = π 4

a ( t ) = 2 2 , 2 , 0

Got questions? Get instant answers now!

Given the vector-valued function r ( t ) = tan t , sec t , 0 (graph is shown here), find the following:

This figure is the graph of a curve in 3 dimensions. The curve is inside of a box. The box represents an octant. The curve has asymptotes that are the diagonals of the box. The curve is hyperbolic.

Speed

v ( t ) = sec 4 t + sec 2 t tan 2 t = sec 2 t ( sec 2 t + tan 2 t )

Got questions? Get instant answers now!

Find the minimum speed of a particle traveling along the curve r ( t ) = t + cos t , t sin t t [ 0 , 2 π ) .

2

Got questions? Get instant answers now!

Given r ( t ) = t i + 2 sin t j + 2 cos t k and u ( t ) = 1 t i + 2 sin t j + 2 cos t k , find the following:

d d t ( r ( t ) × u ( t ) )

0 , 2 sin t ( t 1 t ) 2 cos t ( 1 + 1 t 2 ) , 2 sin t ( 1 + 1 t 2 ) + 2 cos t ( t 2 t )

Got questions? Get instant answers now!

Now, use the product rule for the derivative of the cross product of two vectors and show this result is the same as the answer for the preceding problem.

Got questions? Get instant answers now!

Find the unit tangent vector T (t) for the following vector-valued functions.

r ( t ) = t , 1 t . The graph is shown here:

This figure is the graph of a hyperbolic curve. The y-axis is a vertical asymptote and the x-axis is the horizontal asymptote.

T ( t ) = t 2 t 4 + 1 , −1 t 4 + 1

Got questions? Get instant answers now!

r ( t ) = t cos t , t sin t

Got questions? Get instant answers now!

r ( t ) = t + 1 , 2 t + 1 , 2 t + 2

T ( t ) = 1 3 1 , 2 , 2

Got questions? Get instant answers now!

Evaluate the following integrals:

( e t i + sin t j + 1 2 t 1 k ) d t

Got questions? Get instant answers now!

0 1 r ( t ) d t , where r ( t ) = t 3 , 1 t + 1 , e t

3 4 i + ln ( 2 ) j + ( 1 1 e ) j

Got questions? Get instant answers now!

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask