Trouble understanding vector hat notation - Circular Motion

In summary, the end of the acceleration equation on the slide makes sense when ω is constant and linear speed is also constant.
  • #1
lightlightsup
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9
Homework Statement
See slide 23: http://web.mit.edu/8.01t/www/materials/Presentations/Presentation_W04D1.pdf
Relevant Equations
I don't understand hat notation?
I'm new to classical mechanics.
I've done enough work with vectors to get the basics.
But, I'm having trouble understanding the notation on this MIT presentation I found on circular motion: http://web.mit.edu/8.01t/www/materials/Presentations/Presentation_W04D1.pdf
On slide 23, for example, I don't understand how they keep multiplying (it seems) with r-hat (is that how you say it?).
I don't quite get how to translate the position equation on the top of the slide either.
I'll be happy with either an explanation or a link to good resources to read on the topic.
Thanks in advance!
 

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  • #2
First, yes that's how you say it "r-hat" and "theta-hat".

The basic idea is that ##\hat{r}## and ##\hat{\theta}## are unit vectors, similar to ##\hat{x}## and ##\hat{y}## that you have in Cartesian coordinates. ##\hat{r}## is in the direction of the position vector; and ##\hat{\theta}## is perpendicular to this in a counter-clockwise direction.

There are two big differences between Cartesian and Polar notation:

In Cartesian coordinates, the position vector, ##\vec{r}##, is:

##\vec{r} = x\hat{x} + y\hat{y}##

In polar notation, the position vector involves only ##\hat{r}##. So, we have:

##\vec{r} = r\hat{r}##

Note that ##\hat{\theta}## does not appear in the position vector.

The biggest difference, however, is that the polar unit vectors change with position. Hence, when we are studying motion they change with time. In Cartesian coordinates, once you have chosen your ##x## and ##y## axes the unit vectors are fixed and don't change with position or time.

The velocity vector in Cartesian coordinates is simply:

##\vec{v} = \frac{d \vec{r}}{dt} = \frac{dx}{dt} \hat{x} + \frac{dy}{dt} \hat{y}##

The velocity vector, therefore, is more complicated in polar notation, as you must calculate how the unit vectors change over time:

##\vec{v} = \frac{d \vec{r}}{dt} = \frac{dr}{dt} \hat{r} + r \frac{d \hat r}{dt}##

It turns out that in general ##\frac{d \hat r}{dt} = \frac{d \theta}{dt} \hat \theta## (this is a non-trivial exercise to show this). Putting this into our last equation gives:

##\vec{v} = \frac{dr}{dt} \hat{r} + r \frac{d \theta }{dt} \hat{\theta}##

You can now take the special case of motion in a circle, where ##r## is constant to get:

##\vec{v} = r\frac{d \theta }{dt} \hat{\theta}##

Note: I think this material is quite tricky, so you have to keep working at it.
 
Last edited:
  • #3
Thanks. I definitely need to work on my understanding of vector notation.
But:
On that slide I posted, does the end of the acceleration equation make sense?
When ω is constant, linear speed ν is also constant. Hence, there is no at or α (angular acceleration).
But, there is ar/c which is given by ν2/r or ω2/r.
What I don't understand is why the position vector r or r(t) is repeatedly placed in there like that. It's not being multiplied with right?
Or, is it to be read as "acceleration at that position vector".
 
  • #4
PeroK said:
##\vec{v} = \frac{d \theta }{dt} \hat{\theta}##
Typo: ##\vec{v} =r \frac{d \theta }{dt} \hat{\theta}##
 
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  • #5
lightlightsup said:
Thanks. I definitely need to work on my understanding of vector notation.
But:
On that slide I posted, does the end of the acceleration equation make sense?
When ω is constant, linear speed ν is also constant. Hence, there is no at or α (angular acceleration).
But, there is ar/c which is given by ν2/r or ω2/r.
What I don't understand is why the position vector r or r(t) is repeatedly placed in there like that. It's not being multiplied with right?
Or, is it to be read as "acceleration at that position vector".
It's not just being " placed in there". It arises as part of the differentiation process, according to the rules of calculus.
 
  • #6
lightlightsup said:
Thanks. I definitely need to work on my understanding of vector notation.
But:
On that slide I posted, does the end of the acceleration equation make sense?
When ω is constant, linear speed ν is also constant. Hence, there is no at or α (angular acceleration).
But, there is ar/c which is given by ν2/r or ω2/r.
What I don't understand is why the position vector r or r(t) is repeatedly placed in there like that. It's not being multiplied with right?
Or, is it to be read as "acceleration at that position vector".
In uniform circular motion, I.e. at constant speed, acceleration is in the opposite direction to the position vector.

Acceleration is a vector, so has direction. In general acceleration will have components in both the ##\hat r## and ##\theta## directions.
 
  • #7
Got you. Any good reading material on vectors and vector calculus you can suggest? A book or a website would do.
 
  • #8
lightlightsup said:
Got you. Any good reading material on vectors and vector calculus you can suggest? A book or a website would do.
I like "Paul's Online Maths" for all things calculus.
 
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Related to Trouble understanding vector hat notation - Circular Motion

1. What is vector hat notation?

Vector hat notation is a way of representing vectors in circular motion. It involves placing a small arrow, or "hat," on top of a vector to indicate that it is a unit vector, meaning it has a magnitude of 1.

2. Why is vector hat notation used in circular motion?

In circular motion, the direction of the vector is constantly changing. Vector hat notation allows us to focus on the direction of the vector without having to constantly adjust for changes in magnitude.

3. How do you calculate a unit vector using vector hat notation?

To calculate a unit vector using vector hat notation, you divide the vector by its magnitude. This will give you a new vector with a magnitude of 1 and the same direction as the original vector.

4. What is the purpose of using unit vectors in circular motion?

Unit vectors are useful in circular motion because they allow us to describe the direction of the vector without being affected by changes in magnitude. This makes calculations and analysis of circular motion much simpler and more precise.

5. Are there different types of vector hat notation?

Yes, there are a few variations of vector hat notation, but they all serve the same purpose of representing unit vectors in circular motion. Some common variations include placing a tilde (~) or a circumflex (^) on top of the vector instead of a hat (^).

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