Why is ##2 \pi /T## multiplied by R for v?

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In summary, the conversation discusses the relationship between velocity and angular frequency in a particle moving on a circle with constant angular velocity. The position vector and velocity of the particle are given, and it is shown that the magnitude of the velocity is equal to the product of the angular velocity and the radius. The conversation also clarifies that the first relation mentioned is for angular frequency, not velocity, and that the angular velocity in the example is equal to the angular frequency multiplied by the unit vector in the z-direction.
  • #1
Hey all, $$v = 2 \pi f =2 \pi \frac{1}{T} =\frac{2 \pi }{T} $$ but why is it multiplied by $$R$$? Any help appreciated.
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I see now why R is multiplied in now, but why inst L multiplied in in the analogous pendulum equation?
 
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  • #2
A particle moving with constant angular velocity ##\omega## on a circle is decribed by the position vector
$$\vec{x}=\begin{pmatrix}
r \cos(\omega t) \\ r \sin (\omega t) \\ 0
\end{pmatrix}.$$
The time derivative gives the velocity
$$\dot{\vec{x}}=\vec{v}=\begin{pmatrix}
-r \omega \sin(\omega t) \\ r \omega \cos(\omega t)
\end{pmatrix}.$$
The magnitude thus is
$$|\vec{v}|=r \omega.$$
 
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  • #3
The first relation you write is angular frequency not velocity: ##\omega = \frac{2\pi}{T}##. Then ##\upsilon = \frac{2\pi}{T} R = \omega R##
 
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  • #4
Ok, to be very precise, the angular velocity in my example is ##\vec{\omega}=\omega \vec{e}_z##.
 
  • #5
Thanks all, :)
 

1. Why is ##2 \pi /T## multiplied by R for v?

The factor of ##2 \pi /T## is multiplied by the radius (R) for velocity (v) because it represents the angular velocity of an object in circular motion. This factor, also known as the angular frequency, is the number of complete rotations an object makes per unit time. By multiplying it by the radius, we can calculate the linear speed of the object as it moves along the circumference of the circle.

2. How is the formula ##2 \pi /T## derived?

The formula ##2 \pi /T## is derived from the relationship between angular velocity (ω) and linear velocity (v) in circular motion. The angular velocity is equal to 2π divided by the time it takes for an object to complete one rotation, which is represented by T. This can be written as ω = 2π/T. By substituting v for ωR, where R is the radius, we get v = ωR = (2π/T)R, which simplifies to v = 2πR/T.

3. Why is it important to consider the radius when calculating velocity?

The radius is an important factor to consider when calculating velocity because it affects the distance an object travels in circular motion. The larger the radius, the greater the distance an object will travel in one rotation. Therefore, the radius is used in the formula for velocity to accurately measure the linear speed of an object in circular motion.

4. Does the radius affect the velocity of an object in circular motion?

Yes, the radius does affect the velocity of an object in circular motion. As mentioned before, the larger the radius, the greater the distance an object travels in one rotation. This means that an object with a larger radius will have a higher linear speed compared to an object with a smaller radius, even if they have the same angular velocity.

5. Can the formula ##2 \pi /T## be used for any type of circular motion?

Yes, the formula ##2 \pi /T## can be used for any type of circular motion as long as the object is moving at a constant angular velocity. This includes uniform circular motion, where the object moves at a constant speed along the circumference, and non-uniform circular motion, where the object's angular velocity changes over time. In both cases, the formula can be used to calculate the object's linear speed at any given moment.

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