#$&* course Mth 277 query_10_5*********************************************
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: If V(0) = <5,-2,4> and A(0) = <1,3,-9>, what is A_T and A_N at t = 0? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: a_t =VdotA/sqrt(V) a_t=(-37) / sqrt(25 + 4 + 16) a_t=-37 / sqrt(45) a_t= -37sqrt(5) / 15 a_t=-5.5 a_n=||(v X a)|| / ||v|| a_n=||<5,-2,4> X <1,3,-9>|| / sqrt(45) a_n=||<6, -49, 17>|| / 3sqrt(5) a_n=sqrt(2726) / 3sqrt(5) a_n=7.8 confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: An object moves with a constant angular velocity omega around the circle x^2 + y^2 = r^2 in the xy-plane. Find a parameterization for the circle. Compute the tangential and normal acceleration for the object. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^2 + y^2 = r^2 x=rcos(theta) y=rsin(theta) r(t)=rcos(theta)i + rsin(theta)j v(t)=-rsin(theta)i + rcos(theta)j a(t)=-rcos(theta)i - rsin(theta)j ||v(t)|| = sqrt[r^2sin(theta) + r^2cos(theta)] v(t) dot a(t) / ||v(t)|| = <-rsin(theta)i + rcos(theta)j> dot <-rcos(theta)i - rsin(theta)j> / r = r^2sin(theta)cos(theta) -r^2sin(theta)cos(theta) / r
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating: ********************************************* Question: Consider the vector function R(t) = <3 sin t, 4t, 3 cos t>. Evaluate V(t) = R'(t), N(t), and A(t) = R''(t) when t = 1. Find the vector projection of A(1) onto V(1). Denote this proj_V(1) (A(1)). Find the vector projection of A(1) onto N(1). Denote this proj_N(1) (A(1)). What is the sum of proj_V(1) (A(1)) and proj_N(1) (A(1)). How does proj_V(1) (A(1)) relate to A_T when t = 1. How does proj_N(1) (A(1)) relate to A_N when t = 1. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: V(t)=<3cos(t),4,-3sin(t)> V(1)=<1.62,4,-2.52> A(t)=<-3sin(t),0,-3cos(t)> A(1)=R''(1)=<-2.52,0,-1.62> T(t)=<3cos(t),4,-3sin(t)>/(sqrt((3cos^2(t)+16-3sin^2(t)))) N(t)=T'(t)/||T'(t)|| A(1) onto V(1) =(V(1) dot A(1))V(1)/(||V(1)||^2) proj_V(1) (A(1))=<0,0,0> I believe when you add together these two projections you get R""(1). Since proj_V(1) (A(1))=<0,0,0> proj_N(t)(A(1))=<-2.52,0,-1.62>
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating: ********************************************* Question: Let B = T X N when T and N are the unit tangent and normal vectors to a curve C with position vector R. Show that dB/ds = T X (dN/ds). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: T and N perp unit vect Therefore, there cross product is perp to both vectors a_t=v(t) dot a(t)/||v(t)|| a_n=||v X a||/||v|| confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): This one gets extremely confusing. ------------------------------------------------ Self-critique rating:"