Angular velocity and vectors

[wizard85]

Angular velocity and vectors...

Homework Statement

Given the following vectors:

$$\vec A=\hat i +\hat j - 2\hat k$$ and $$\vec C=\hat j - 5\hat k$$

Let $$\vec A$$ and $$\vec C$$ be drown from a common origin and let $$\vec C$$ rotate about $$\vec A$$ with angular velocity $$\vec w$$ of $$2 \frac{rad}{s}$$. Find the velocity $$\vec v$$ of the head of $$\vec C$$.

Homework Equations

The Attempt at a Solution

My step-by-step way for resolving it, is:

1)I know that $$\vec v= w \times \vec C$$
2) By multiplying: $$\vec A \times \vec C$$ I'll find a vector parallel to $$\vec w$$ namely D
3) Now, $$\vec D= \vec A\times \vec C=(\hat i +\hat j - 2\hat k) \times (\hat j - 5\hat k) = 7*\hat i -5*\hat j +\hat k$$

4) I also know that $$\vec w$$ is obtained by a linear combination of $$\vec D$$'s parameter. Then:

$$\vec w= a * \vec D=a * (7*\hat i -5*\hat j +\hat k)$$

but $$|\vec w|= 2$$ so $$a= \frac{2}{|\vec D|}$$ --> $$a=\sqrt{75}$$. Finally $$\vect w= \frac{2}{\sqrt{75}} (7*\hat i -5*\hat j +\hat k)$$

Thus:

$$\vect v= \vect w \times \vect C = \frac{2}{\sqrt{75}} (7*\hat i -5*\hat j +\hat k) \times (\hat j - 5\hat k)$$

is that correct?

Thanks to all...

 

[wizard85]

Homework Statement

Given the following vectors:

$$\vec A=\hat i +\hat j - 2\hat k$$ and $$\vec C=\hat j - 5\hat k$$

Let $$\vec A$$ and $$\vec C$$ be drown from a common origin and let $$\vec C$$ rotate about $$\vec A$$ with angular velocity $$\vec w$$ of $$2 \frac{rad}{s}$$. Find the velocity $$\vec v$$ of the head of $$\vec C$$.

Homework Equations

The Attempt at a Solution

My step-by-step way for resolving it, is:

1)I know that $$\vec v= w \times \vec C$$
2) By multiplying: $$\vec A \times \vec C$$ I'll find a vector parallel to $$\vec w$$ namely D
3) Now, $$\vec D= \vec A\times \vec C=(\hat i +\hat j - 2\hat k) \times (\hat j - 5\hat k) = 7*\hat i -5*\hat j +\hat k$$

4) I also know that $$\vec w$$ is obtained by a linear combination of $$\vec D$$'s parameter. Then:

$$\vec w= a * \vec D=a * (7*\hat i -5*\hat j +\hat k)$$

but $$|\vec w|= 2$$ so $$a= \frac{2}{|\vec D|}$$ --> $$a=\sqrt{75}$$. Finally $$\vect w= \frac{2}{\sqrt{75}} (7*\hat i -5*\hat j +\hat k)$$

Thus:

$$\vect v= \vect w \times \vect C = \frac{2}{\sqrt{75}} (7*\hat i -5*\hat j +\hat k) \times (\hat j - 5\hat k)$$

is that correct?

Thanks to all...

nobody?

 

[thomate1]

I can confirm that your steps are correct in finding the velocity vector of the head of \vec C. You have correctly used the formula \vec v= w \times \vec C and found the vector \vec D parallel to \vec w. Your final result of \vect v= \frac{2}{\sqrt{75}} (7*\hat i -5*\hat j +\hat k) \times (\hat j - 5\hat k) is also correct.

In general, angular velocity and vectors are important concepts in physics and are often used to describe rotational motion. Angular velocity is a vector quantity that describes the rate of change of angular displacement with respect to time, and it is typically measured in radians per second. Vectors are used to represent quantities that have both magnitude and direction, and they are an essential tool for analyzing motion in physics.

In this problem, we are given two vectors \vec A and \vec C and asked to find the velocity vector of the head of \vec C as it rotates about \vec A with a given angular velocity \vec w. By using the formula \vec v= w \times \vec C, we can find the velocity vector as a result of the rotation. This problem also demonstrates the use of vector multiplication, specifically the cross product, to find a vector that is parallel to \vec w.

Overall, this problem highlights the importance of understanding both angular velocity and vectors in order to accurately describe and analyze rotational motion.