Determining unit vector of acceleration and velocity in circular motio

In summary, the unit vector for velocity in circular motion is the velocity vector divided by its magnitude, which is the same as for any other motion. An example of this can be seen in circular motion around the origin at a constant speed and radius, where the unit vector for position is the sine and cosine components in Cartesian coordinates.
  • #1
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As a particle orbits around a circle, the unit vector of the velocity and acceleration component is constantly changing, so, how do I determine the unit vector?
 
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  • #2
The unit vector for, say, velocity, is the velocity vector divided by the vector-magnitude - same as for any motion.
What is the problem? Can you provide an example where a difficulty arises?

$$\vec v = v\hat v: \hat v = \frac{\vec v}{v}$$

eg. Circular motion about origin at constant speed v and radius R, in Cartesian coordinates:

##\vec r (t) = \hat\imath R\sin\omega t + \hat\jmath R\cos\omega t : v=R\omega##

The unit vector for position would be:

##\hat r = \hat\imath \sin\omega t + \hat\jmath \cos\omega t ##
 
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  • #3
Simon Bridge said:
The unit vector for, say, velocity, is the velocity vector divided by the vector-magnitude - same as for any motion.
What is the problem? Can you provide an example where a difficulty arises?

$$\vec v = v\hat v: \hat v = \frac{\vec v}{v}$$

eg. Circular motion about origin at constant speed v and radius R, in Cartesian coordinates:

##\vec r (t) = \hat\imath R\sin\omega t + \hat\jmath R\cos\omega t : v=R\omega##

The unit vector for position would be:

##\hat r = \hat\imath \sin\omega t + \hat\jmath \cos\omega t ##

Just as I suspected. I wanted to confirm my understanding.
Thanks for the input.
 
  • #4
Gah - I think I got suckered: well done!
In future - if you want to confirm your understanding, just state your understanding and ask.
 
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  • #5


To determine the unit vector of velocity and acceleration in circular motion, we must first understand the nature of circular motion. In circular motion, the velocity and acceleration vectors are always perpendicular to each other, with the velocity vector tangent to the circle and the acceleration vector pointing towards the center of the circle.

To find the unit vector of velocity, we can use the formula v = ωr, where v is the velocity, ω is the angular velocity, and r is the radius of the circle. This formula tells us that the magnitude of the velocity is equal to the product of the angular velocity and the radius. To find the direction of the velocity vector, we can use the right-hand rule, where the direction of the vector is determined by curling the fingers of the right hand in the direction of the rotation and the thumb will point in the direction of the velocity vector.

To find the unit vector of acceleration, we can use the formula a = ω²r, where a is the acceleration, ω is the angular velocity, and r is the radius of the circle. This formula tells us that the magnitude of the acceleration is equal to the product of the square of the angular velocity and the radius. The direction of the acceleration vector is always towards the center of the circle, so we can use the same right-hand rule to determine its direction.

In summary, to determine the unit vector of velocity and acceleration in circular motion, we can use the formulas v = ωr and a = ω²r and the right-hand rule to determine their magnitudes and directions. It is important to note that in circular motion, the unit vectors of velocity and acceleration are constantly changing as the particle moves around the circle, so it is crucial to consider the specific position and direction of the particle at any given moment in order to accurately determine the unit vectors.
 

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is used to represent a direction in space.

2. How is the unit vector of acceleration determined in circular motion?

The unit vector of acceleration in circular motion is determined by dividing the acceleration vector by its magnitude, resulting in a vector with a magnitude of 1 pointing in the same direction as the original acceleration vector.

3. What is the unit vector of velocity in circular motion?

The unit vector of velocity in circular motion is a vector with a magnitude of 1 pointing in the direction of the tangent to the circular path at a specific point.

4. How is the unit vector of velocity calculated in circular motion?

The unit vector of velocity in circular motion is calculated by dividing the velocity vector by its magnitude, resulting in a vector with a magnitude of 1 pointing in the same direction as the original velocity vector.

5. Why is it important to determine the unit vectors of acceleration and velocity in circular motion?

Determining the unit vectors of acceleration and velocity in circular motion allows us to understand the direction of these vectors and their relationship to the circular path. This information is crucial in analyzing circular motion and can help us make predictions and calculations about the motion.

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