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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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.
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
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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