Answer:
54.8 Nm
Step-by-step explanation:
Torque is the cross product of the radius vector and force vector:
τ = r × F
Another way to write it is the product of the radius and force magnitudes times the sine of the angle between the vectors.
τ = rF sin θ
Here, r = 0.366 meters and F = 155 Newtons. F is in the +y direction, and r is 15° below the +x axis, so the angle between the vectors is 90° − 15° = 75°.
τ = (0.366 m) (155 N) (sin 75°)
τ = 54.8 Nm
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Answer:
710%
Step-by-step explanation:
1 bucket of eggs is charged at $16.20 for 450 eggs.
1 dozen = 12
450 = 450/12 = 37.5
The store pays ($16.20/37.5 doz)=$0.432/doz
The store charges ($3.50-0.432) $3.068 more per dozen than they paid.
($3.068/$0.432)(100%) = 710%
Answer:
6.99
Step-by-step explanation:
Answer:
10
Step-by-step explanation:
We require to find the cube root (n = 3) pf a = 1000, that is
the root is
= 10 ( since 10 × 10 × 10 = 1000 )
Stokes' theorem equates the line integral of
along the curve to the surface integral of the curl of
over any surface with the given curve as its boundary. The simplest such surface is the triangle with vertices (1,0,1), (0,1,0), and (0,0,1).
Parameterize this triangle (call it
) by
![\vec s(u,v)=(1-v)((1-u)(1,0,1)+u(0,1,0))+v(0,0,1)](https://tex.z-dn.net/?f=%5Cvec%20s%28u%2Cv%29%3D%281-v%29%28%281-u%29%281%2C0%2C1%29%2Bu%280%2C1%2C0%29%29%2Bv%280%2C0%2C1%29)
![\vec s(u,v)=((1-u)(1-v),u(1-v),1-u+uv)](https://tex.z-dn.net/?f=%5Cvec%20s%28u%2Cv%29%3D%28%281-u%29%281-v%29%2Cu%281-v%29%2C1-u%2Buv%29)
with
and
. Take the normal vector to
to be
![\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}=(0,1-v,1-v)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%3D%280%2C1-v%2C1-v%29)
Divide this vector by its norm to get the unit normal vector. Note that this assumes a "positive" orientation, so that the boundary of
is traversed in the counterclockwise direction when viewed from above.
Compute the curl of
:
![\vec F=(2x,2y,2x+2z)\implies\mathrm{curl}\vec F=(0,-2,0)](https://tex.z-dn.net/?f=%5Cvec%20F%3D%282x%2C2y%2C2x%2B2z%29%5Cimplies%5Cmathrm%7Bcurl%7D%5Cvec%20F%3D%280%2C-2%2C0%29)
Then by Stokes' theorem,
![\displaystyle\int_{\partial T}\vec F\cdot\mathrm d\vec r=\iint_T\mathrm{curl}\vec F\cdot\mathrm d\vec S](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_%7B%5Cpartial%20T%7D%5Cvec%20F%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Ciint_T%5Cmathrm%7Bcurl%7D%5Cvec%20F%5Ccdot%5Cmathrm%20d%5Cvec%20S)
where
![\mathrm d\vec S=\dfrac{\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}}{\left\|\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}\right\|}\,\mathrm dS](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cvec%20S%3D%5Cdfrac%7B%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%7D%7B%5Cleft%5C%7C%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%7D%5C%2C%5Cmathrm%20dS)
![\mathrm d\vec S=\dfrac{\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}}{\left\|\frac{\partial\vec s}{\partial u}\times\frac{\partial\vec s}{\partial v}\right\|}\left\|\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right\|\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cvec%20S%3D%5Cdfrac%7B%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%7D%7B%5Cleft%5C%7C%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%7D%5Cleft%5C%7C%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![\mathrm d\vec S=\left(\dfrac{\partial\vec s}{\partial u}\times\dfrac{\partial\vec s}{\partial v}\right)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cmathrm%20d%5Cvec%20S%3D%5Cleft%28%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cvec%20s%7D%7B%5Cpartial%20v%7D%5Cright%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
The integral thus reduces to
![\displaystyle\int_0^1\int_0^1(0,-2,0)\cdot(0,1-v,1-v)\,\mathrm du\,\mathrm dv=\int_0^12(v-1)\,\mathrm dv=\boxed{-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_0%5E1%5Cint_0%5E1%280%2C-2%2C0%29%5Ccdot%280%2C1-v%2C1-v%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv%3D%5Cint_0%5E12%28v-1%29%5C%2C%5Cmathrm%20dv%3D%5Cboxed%7B-1%7D)