Question

Find T​, N​, and kappa for the plane curve Bold r left parenthesis t right parenthesis equalsleft parenthesis 7 Bold cos t plus 7 t Bold font size decreased by 1 sin t right parenthesis Bold i plus left parenthesis 7 Bold sin t minus 7 t Bold font size decreased by 1 cos t right parenthesis Bold j​, t greater than 0 .

Find T, N, and for the plane curve r(t) = (7 cost + 7t sin t)i + (7 sin t - 7t cos t)j, t> 0.

Answer 1

Explanation:

r(t) = (7 cost + 7t sin t)i + (7 sin t - 7t cos t)j

`(d \bar r t)/(dt) =(7(d)/(dt)\cos t + 7(d)/(dt) (t \sin t)i+(7(d)/(dt) \sin t-7(d)/(dt) t \cos t)j`

`=(7(-\sin t)+7(1* \sin t+t \cos t))i+(7 \cost -7(1*\cos t - t \sin t))j\\\\=7((-\sin t+\sin t+t \cos t)i+(\cos t-\cos t+t \sin t)j)\\\\=7((t\cos t)i+(t\sin t)j)`

`\bar r'(t)=(d \bar r t)/(dt) =(7t\cos t)i+(7t\sin t)j---(1)\\\\11\bar r(t)=√((7t\cos t)^2+(7t\sin t)^2)\\\\=√(49t^2(\cos^2t+\sin^2 t)) \\\\=7t`

`\bar T (t)=(\bar r'(t))/(11\bar r(t)11) =((7t\cos t)i+(7t\sin t)j)/(7t) \\\\\barT(t)=(\cos t)i+(\sin t)j`

`\bar T'(t)=(d)/(dt) (\cos t)i+(d)/(dt) (\sin t) j\\\\\bar T'(t)=(-\sin t)i+(\cos t)j---(2)\\\\11\bar T'(t)=√((-\sin t)^2+(\cos t)^2) \\\\=√(\sin^2t+\cos^2t) \\\\=1`

`\bar N(t)=\bar T'(t)=((-\sin t)i+(\cos t)j)/((1)) \\\\ \large \boxed {\bar N(t)=(-\sin t)i+(\cos t)j}`

`K(t)=(|\b\r T'(t)|)/(\bar r (t)|) \\\\=(|-\sin t i+\cos t j|)/(|7t\cos t +7t \sin t j|)`

Using eq (1) and (2)

`K(t)=(√((-\sin t)^2+(\cos t)^2) )/(√((7t\cos t)^2+(7t\sin t)^2) )\\\\=(√(\sin^2 t+\cos^2t) )/(√(49t^2(\cos^2 t+\sin^2t)) )\\\\=(√(1) )/(√(49t^2* 1) ) \\\\ \large \boxed {K(t)=(1)/(7t) }`