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_5

No q_a_ appears to be necessary for this section.  You may proceed to the section and the problem assignment.

The problem assignment is repeated here for your convenience:

Section 10.5

1) Find the tangential and normal components of an object's acceleration which has the position vector R(t) = <2 cos t, 5 sin t>.

2) Find the tangential and normal components of an object's acceleration which has the position vector R(t) = <3/5 cos t, 4/5(1+sin t), cos t>.

3) If V(0) = -2i - 3j and A(0) = 2i + 3j, what is A_T and A_N at t = 0?

4) If V(0) = <5,-2,4> and A(0) = <1,3,-9>, what is A_T and A_N at t = 0?

5) ||V|| = sqrt(e^-t + t^4). What is A_T at t = 0?

6) Where on the trajectory of R(t) = <2t^2 - 5t, 5t + 2, 4t^2> is the speed maximized and minimized?

7) 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.

8) A car which weighs 3,000lbs moves along the elliptic path 1600x^2 + 100y^2 = 1, where x and y are measured in miles. If the car travels at the constant speed of 30mi/hr, how much frictional force is required to keep it from skidding as it turns at (1/40,0)? What about at (0,1/10)?

9) ** 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.

10) ** Use the method described in the previous problem to find A_T and A_N for R(t) = <cos t, sin t>.

11) ** 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).

12) An object connected to a string of length r is spun counterclockwise in a circular path in a horizontal plane. Let omega be the constant angular velocity of the object.

• Show that the position vector of the object is R(t) = (r cos omega t) i + (r sin omega t) j.
• Find the normal component of the object's acceleration.
• If omega is doubled, what is the effect on the normal component of acceleration?
• What is the effect on A if omega is unchanged but the length of the string is doubled?
• What happens to A if omega is doubled but r is halved?