Calculating Rotational Motion: Proving Velocity is Perpendicular to Radius

In summary, the conversation discusses the use of cylindrical coordinates to describe circular motion and how to show that the velocity is perpendicular to the radius in this type of motion. The use of cylindrical coordinates is only advantageous if the circular motion is centered around the z-axis. Conditions must be set, such as \dot{p} = \dot{z} = 0, to deduce something about circular motion.
  • #1
Phymath
184
0
I calculated the equations for circlular motion as fallows...

[tex]
\vec{r}= p \hat{e_p} + z \hat{e_z}[/tex] where e_p = unit vector in the radial direction, and so on

[tex]
\frac{\partial{\vec{r}}}{\partial{t}} = \dot{p}\hat{e_p} + p\dot{\theta}\hat{e_{\theta}} + \dot{z}\hat{e_z} [/tex]

how do i show that the velocity is perpendicular to the radius, show me my mistake, but...
[tex] \vec{r} \bullet \vec{\dot{r}} = p \dot{p} + z \dot{z} [/tex] which isn't obvious to me that that is 0 so what to do?
 
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  • #2
Maybe I'm missing the point of what you are doing, but if you want to deduce something about circular motion, you must impose some conditions. What you've described so far is just the use of cylindrical coordinates to describe any motion. For circular motion, set [itex]\dot{p} = \dot{z} = 0[/itex]; then all terms in the velocity drop out except [itex]p\dot{\theta}\hat{e_{\theta}}[/itex].
 
  • #3
i agree the dp/dt would be zero, however if circle is oriented at some angle to a xyz coord system, then wouldn't dz/dt be non-zero

bare with the picture the blue lines are there to help u reference where in the plane the point is
 

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  • #4
Phymath said:
i agree the dp/dt would be zero, however if circle is oriented at some angle to a xyz coord system, then wouldn't dz/dt be non-zero
Sure. But if your circle is oriented at an angle to your z-axis, you've chosen an inconvenient coordinate system to describe the circular motion. The advantage of cylindrical coordinates is only apparent if the circular motion is centered around the z-axis.
 
  • #5
i agree it is inconvenient however should it not be able to be calculated, not sure what u mean bye the "advantage of cylindrical coordinates is only apparent if the circluar motion is centered around the z-axis"
 

What is rotational motion?

Rotational motion is the movement of an object around an axis or center point. It is also known as circular motion.

Why is it important to prove that velocity is perpendicular to radius in rotational motion?

Proving that velocity is perpendicular to radius in rotational motion is important because it helps us understand the direction and magnitude of an object's motion. It also allows us to calculate important quantities, such as angular velocity and centripetal acceleration.

How do you calculate rotational motion?

To calculate rotational motion, you need to know the object's angular velocity, radius, and linear velocity. The formula for rotational motion is ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius.

What is the relationship between velocity and radius in rotational motion?

In rotational motion, velocity and radius are always perpendicular to each other. This means that the direction of the object's velocity is always tangent to the circle it is moving in, while the radius is always perpendicular to the tangent line.

How can you prove that velocity is perpendicular to radius in rotational motion?

You can prove that velocity is perpendicular to radius in rotational motion using mathematical proofs or by conducting experiments. One way to prove this is by using the concept of centripetal acceleration, which is always directed towards the center of the circle and is perpendicular to the object's velocity.

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