Finding unit normal vectors and normal/tangent components of accelerat

In summary, the student is trying to find a solution to a homework equation involving the cost, sint, and t coordinates. They are looking for a vector that is simply differentiated and uses identities. The student is having trouble with simplifying the unit tangent vector and is looking for a way to do it quickly.
  • #1
icesalmon
270
13

Homework Statement


given r(t) = <t, 1/t,0> find T(t) N(t) aT and aN at t = 1


Homework Equations


T(t) = r'/||r'||
N(t) = T'/||T'||
aT = a . T = (v . a)/||T||
aN = a . N = ||v x a||/||v|| = sqrt(||a||2 - aT2)


The Attempt at a Solution


for my T(t) I get <t2, -1 , 0>/(sqrt(1+t4) (I like keeping things in 3 dimensions even if there is no contribution in the z direction)
and I am not calculating the normal vector if there isn't some algebra I can use to simplify this greatly , which I am not seeing, t4 + 1 I don't believe I can factor and I can't think of any other way to simplify this one so I'm just moving around through the problem set looking for some r(t) = <cost, sint, t> type of vector that I can simply differentiate and use identities with, desperately trying to avoid those other problems
 
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  • #2
icesalmon said:

Homework Statement


given r(t) = <t, 1/t,0> find T(t) N(t) aT and aN at t = 1


Homework Equations


T(t) = r'/||r'||
N(t) = T'/||T'||
aT = a . T = (v . a)/||T||
aN = a . N = ||v x a||/||v|| = sqrt(||a||2 - aT2)


The Attempt at a Solution


for my T(t) I get <t2, -1 , 0>/(sqrt(1+t4)
How did you get that? Your first equation in the "relevant equations" section is ##T(t) = r' / \|r\|##. What is ##r'## in this case?
 
  • #3
r' = <1,-1/t2>
|r'| = sqrt(1 + 1/t4)
did I mess something up when differentiating/doing algebra/both?
 
  • #4
|r'| = sqrt((t4+1)/t4) = (1/t2)sqrt(t4+1)
which should be t2r'/(|r'|)
which I believe is actually <t2,-1>/(sqrt(t4+1)
which is the same thing...oops I'm not sure what I'm messing up here.
 
  • #5
icesalmon said:
|r'| = sqrt((t4+1)/t4) = (1/t2)sqrt(t4+1)
which should be t2r'/(|r'|)
which I believe is actually <t2,-1>/(sqrt(t4+1)
which is the same thing...oops I'm not sure what I'm messing up here.
OK, that looks fine. So what is ##T'##? Just use the quotient rule, it shouldn't be too horrible.
 
  • #6
well that's what I'm saying, are there any ways I can simplify this particular unit tangent vector before differentiating? I know how to differentiate it, but on an exam I will chew up a ton of time making sure I keep everything straight with the differentiation in terms of signs and cancellations.
 

1. What is a unit normal vector?

A unit normal vector is a vector that is perpendicular to a given surface or curve at a specific point, and has a magnitude of 1. It is used to determine the direction of the normal force, which is the force that is perpendicular to the surface or curve at that point.

2. How is a unit normal vector calculated?

To calculate a unit normal vector, you first need to find the partial derivatives of the surface or curve at the given point. Then, using these partial derivatives, you can calculate the vector by taking the cross product of the two partial derivatives. Finally, divide the resulting vector by its magnitude to get a unit vector.

3. What is the difference between a normal component and a tangent component of acceleration?

The normal component of acceleration refers to the acceleration in the direction of the unit normal vector, while the tangent component of acceleration refers to the acceleration in the direction of the tangent vector. The normal component is responsible for the change in direction of the velocity, while the tangent component is responsible for the change in magnitude of the velocity.

4. Why is finding unit normal vectors and normal/tangent components of acceleration important?

Understanding the unit normal vector and the normal/tangent components of acceleration is important in physics and engineering, as it allows us to analyze the motion of objects and understand the forces acting on them. It also helps in solving problems involving curved motion and determining the normal force, which is a crucial concept in mechanics.

5. Can unit normal vectors and normal/tangent components of acceleration be applied to real-world scenarios?

Yes, these concepts can be applied to real-world scenarios. For example, in roller coaster design, engineers use the normal and tangent components of acceleration to ensure the safety of the riders and to create thrilling experiences. These concepts are also used in sports, such as in analyzing the trajectory of a ball in a curved motion, or in understanding the forces acting on a gymnast during a routine.

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