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This figure has a vector r from an “axis of rotation”. At the terminal point of r there is a vector labeled “F”. The angle between r and F is theta.
Torque measures how a force causes an object to rotate.

Think about using a wrench to tighten a bolt. The torque τ applied to the bolt depends on how hard we push the wrench (force) and how far up the handle we apply the force (distance). The torque increases with a greater force on the wrench at a greater distance from the bolt. Common units of torque are the newton-meter or foot-pound. Although torque is dimensionally equivalent to work (it has the same units), the two concepts are distinct. Torque is used specifically in the context of rotation, whereas work typically involves motion along a line.

Evaluating torque

A bolt is tightened by applying a force of 6 N to a 0.15-m wrench ( [link] ). The angle between the wrench and the force vector is 40 ° . Find the magnitude of the torque about the center of the bolt. Round the answer to two decimal places.

This figure is the image of an open-end wrench. The length of the wrench is labeled “0.15 m.” The angle the wrench makes with a vertical vector is 40 degrees. The vector is labeled with “6 N.”
Torque describes the twisting action of the wrench.

Substitute the given information into the equation defining torque:

τ = r × F = r F sin θ = ( 0.15 m ) ( 6 N ) sin 40 ° 0.58 N · m .
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Calculate the force required to produce 15 N · m torque at an angle of 30 º from a 150-cm rod.

20 N

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Key concepts

  • The cross product u × v of two vectors u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 is a vector orthogonal to both u and v . Its length is given by u × v = u · v · sin θ , where θ is the angle between u and v . Its direction is given by the right-hand rule.
  • The algebraic formula for calculating the cross product of two vectors,
    u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 , is
    u × v = ( u 2 v 3 u 3 v 2 ) i ( u 1 v 3 u 3 v 1 ) j + ( u 1 v 2 u 2 v 1 ) k .
  • The cross product satisfies the following properties for vectors u , v , and w , and scalar c :
    • u × v = ( v × u )
    • u × ( v + w ) = u × v + u × w
    • c ( u × v ) = ( c u ) × v = u × ( c v )
    • u × 0 = 0 × u = 0
    • v × v = 0
    • u · ( v × w ) = ( u × v ) · w
  • The cross product of vectors u = u 1 , u 2 , u 3 and v = v 1 , v 2 , v 3 is the determinant | i j k u 1 u 2 u 3 v 1 v 2 v 3 | .
  • If vectors u and v form adjacent sides of a parallelogram, then the area of the parallelogram is given by u × v .
  • The triple scalar product of vectors u , v , and w is u · ( v × w ) .
  • The volume of a parallelepiped with adjacent edges given by vectors u , v , and w is V = | u · ( v × w ) | .
  • If the triple scalar product of vectors u , v , and w is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
  • The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
  • Torque τ measures the tendency of a force to produce rotation about an axis of rotation. If force F is acting at a distance r from the axis, then torque is equal to the cross product of r and F : τ = r × F .

Key equations

  • The cross product of two vectors in terms of the unit vectors
    u × v = ( u 2 v 3 u 3 v 2 ) i ( u 1 v 3 u 3 v 1 ) j + ( u 1 v 2 u 2 v 1 ) k

For the following exercises, the vectors u and v are given.

  1. Find the cross product u × v of the vectors u and v . Express the answer in component form.
  2. Sketch the vectors u , v , and u × v .

u = 2 , 0 , 0 , v = 2 , 2 , 0

a. u × v = 0 , 0 , 4 ;
This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”

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u = 3 , 2 , −1 , v = 1 , 1 , 0

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u = 2 i + 3 j , v = j + 2 k

a. u × v = 6 , −4 , 2 ;
This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components <2, 3, 0>. The second vector is labeled v and has components <0, 1, 2>.” The third vector is labeled u cross v and has components <6, -4, 2>.”

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u = 2 j + 3 k , v = 3 i + k

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Simplify ( i × i 2 i × j 4 i × k + 3 j × k ) × i .

−2 j 4 k

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Simplify j × ( k × j + 2 j × i 3 j × j + 5 i × k ) .

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In the following exercises, vectors u and v are given. Find unit vector w in the direction of the cross product vector u × v . Express your answer using standard unit vectors.

Questions & Answers

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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Practice Key Terms 6

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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