assignment 9

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course MTH 277

7/1/13

Note that ` in front of a symbol indicates that the symbol is a vector. The only exception: `d means 'Delta'. I will eventually search-replace the document to convert the notation to boldface.If `R(t) = sin(t) `i + cos(`t) j + t `k then:

`q001. What are the associated velocity and acceleration vectors?

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Your solution:

V(t)= cos(t) I - sin(t)j +1 k

A(t)=-sin(t)I -cos(t) j +0 k

confidence rating #$&*:

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Given Solution:

`v(t) = `R ' (t) = -cos(t) `i + sin(t) `j + `k

`a(t) = `v ' (t) = `r '' (t) = sin(t) `i + cos(t) `j

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q002. What is the function describing the unit tangent vector?

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Your solution:

V(t)= cos(t) I - sin(t)j +1 k

Magnitude Sqrt(cos^2(t)+1+sin^2(t)=sqrt(2)

V(t)/mag= answer

confidence rating #$&*:

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Given Solution:

Divide `v(t) by || `v(t) || and simplify

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q003. What is the component of the acceleration vector in the direction of the unit tangent vector?

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Your solution:

V(t)/magnitude*a(t)=

Atan(t)=(-sin(t)I -cos(t) j +0 k)V(t)/v magnitude

confidence rating #$&*:

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Given Solution:

The component is denoted `a_T (t) . The desired component is the projection of `a(t) on `T(t).

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Self-critique (if necessary):

ok

@&

You have the gist of the solution but you haven't provided the specifics. You can find v(t) and || v(t) ||, incorporate them into your expression and simplify.

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Self-critique rating:ok

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Question: `q004. What is the component of the acceleration vector in the direction perpendicular to the unit tangent vector?

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Your solution:

A(t)=Atan+Anormal

A(t)-Atan=anormal

Anorm=-(-sin(t)I -cos(t) j +0 k)V(t)/v magnitude+ sin(t)I -cos(t) j +0 k

confidence rating #$&*:

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Given Solution:

Subtract the component `a_T(t) from `a(t).

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q005. What is the normal component of the acceleration?

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Your solution:

Same as the last one.

confidence rating #$&*:

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Given Solution:

This is the component perpendicular to the unit tangent vector.

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q006. Show that the normal component of the acceleration is perpendicular to the tangential component.

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Your solution: The dot products equal 0.

confidence rating #$&*:

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Given Solution:

Two vectors are perpendicular if their dot product is zero.

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q007. Show that the direction of the derivative of the unit tangent vector is the same as that of the unit normal vector.

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Your solution: when the angle is 0.

confidence rating #$&*:

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Given Solution:

Two vectors are parallel if the cosine of the angle between them is zero.

How therefore can to test to see if the vectors are parallel?

What further test allows us to determine if they are in the same direction, vs. in the opposite directions.

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Self-critique (if necessary):ok

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Self-critique rating:

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Question: `q008. Find the unit normal vector.

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Your solution:

=-(-sin(t)I -cos(t) j +0 k)V(t)/v magnitude+ sin(t)I -cos(t) j +0 k

confidence rating #$&*:

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Given Solution:

You have at least one vector in the normal direction (in fact in the preceding questions you have found two). Use either to find the unit normal.

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q009. Find the unit binormal vector.

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Your solution:

-(-sin(t)I -cos(t) j +0 k)V(t)/v magnitude+ sin(t)I -cos(t) j +0 k + cos(t) I - sin(t)j +1 k /sqrt(2)

confidence rating #$&*:

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Given Solution:

You should have the unit normal and unit tangent. Use them to easily find the unit binormal. How do you know that your result is a unit vector?

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q009. What difference would it make in the above results if the function was `R(t) = sin(t^2) `i + cos(t^2) `j + t `k?

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Your solution:

There would be chain rule to do in the deriviatives.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

ok

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Self-critique rating:ok

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Question: `q010. What difference would it make in the above results if the function was `R(t) = sin(t^2) `i + cos(t^2) `j + t^2 `k?

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Your solution:

The z component would exist even in acceleration .

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):ok

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Self-critique rating:

ok

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Self-critique (if necessary):

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Self-critique rating:

@&

You could give much more complete and specific results, and it may be for lack of time that you haven't gone into all the details here.

However your answers, as far as they go, are generally consistent with a correct solution.

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Do begin doing your work in a text editor. A word process does a lot of things you don't want in your work, for example confusing cases of variables by capitalizing I's and the first letters of paragraphs.

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