If your solution to stated problem does not match the given solution, you should self-critique per instructions at

 

   http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution.  If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.  This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

qa 10_04

The unit tangent vector is the unit vector in the direction of the velocity.

The tangential component of the acceleration is the projection of the acceleration vector on the unit tangent vector.

The normal component of the acceleration is the acceleration, minus its component in the direction of the unit tangent vector. 

The unit normal vector is the unit vector in the direction of the normal component of the acceleration vector.  A vector parallel to the normal vector can also be obtained by taking the derivative of the unit tangent vector.

The unit binormal vector is the cross product of the unit tangent and unit normal vectors.

The speed of a point whose position function is R(t) is the magnitude of the velocity vector.

Section 10.4

... A point moves around a circle of radius A at constant speed v.  If the radial line vector from the center of the circle to the point

The position vector of a moving point is `R(t) = A cos(omega * t) `i + A sin(omega * t) `j.  For this position function:

Question: `q001.  Sketch the path of the point.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q002.  What are the corresponding velocity and acceleration vectors?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q003.  What is the speed of the point and what is the magnitude of the acceleration?  Does either change with respect to t?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

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Question:

`q004.  What is the angle between `R(t) and the velocity vector, and what is the angle between the velocity vector and the acceleration vector? 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q005.  Sketch the position vector, the velocity vector and the acceleration vector at the instant when omega * t = pi / 6.

 

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Given Solution: 

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Question:

`q006.  Let v be the speed of the point and r the radius of the circle.  What is the expression for the magnitude of the acceleration in terms of v and r?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q007.  If another point is moving around a different circle at a different constant speed v, then if the magnitude of the acceleration of that point is a_cent, what is the radius of the circle?

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

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Question:

The position vector of a moving point is `R(t) = A cos(omega * t^2) `i + A sin(omega * t^2) `j. 

`q008.  Sketch the path of the point.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q009.  Find the acceleration and velocity vectors, and the magnitudes of both.  Does either change with respect to t?

 

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Given Solution: 

Self-critique (if necessary):

Question:

`q010.  Find the components of the acceleration in the direction of the velocity, and perpendicular to the direction of the velocity.  Find the magnitudes of both components of the acceleration.

 

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Given Solution: 

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Question:

`q011.  If an object was moving around a circle with constant speed equal to that of this particle, with acceleration toward the center of the circle equal in magnitude to the perpendicular component of the acceleration, what would be the expression for the radius of that circle?  Of course we know that the circle in this example has radius A, but don't use that knowledge in your solution. 

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

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Question:

`q012.  For any position function `R(t), we can follow the same procedure to find the radius of a hypothetical circle.

At any t, we can sketch the osculating circle, which is a circle whose radius is given by the expression found in the preceding question, subject to the condition that the direction of the radial vector (i.e., the vector from the center of that circle to the moving point) is in the direction opposite the unit normal.  This is easily understood from a sketch.  The center and radius of the osculating circle. The path of the particle at The curvature of the path at that instant is equal to the reciprocal of the radius of the osculating circle. 

The osculating circle is the circle of the radius you found in the preceding, whose center

`q013.  The curvature of the path of the particle is equal to the reciprocal of the radius of the osculating circle.

 

Your solution: 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution: 

Self-critique (if necessary):

Question: