Can the values found for Angular Speed/Frequency be substituted for Angular Velocity?

Question

Specificaly in the formula for determing Rotational Energy : 1/2Iw^2,
Where I = Moment of inertia and w = Angular Velocity.

Wikipedia says that angular velocity is used to determine Angular Speed/Frequency, but the formulas are VASTLY different for each, so would they even result in the same unit of measure let alone, the same values?

The Angular Velocity formulas look like chinese to me and seem to0 complicated for the application of just determining energy of rotation. Angular Speed formula looks more like it, but thats not what the Rotational Energy formula calls for, is Wikipedia wrong or are they interchangable?

is there a different formula for Rotational Energy that utilized rotational speed?

Answer 1

Angular velocity (w) is expressed in radians per second. 2 * pi radians = 360 degrees = 1 revolution = 1 cycle.

The trick is expressing Angular speed/frequency in terms of radians per second (w using your notation).

Ex: Angular speed of 300 RPM is:

300 revolutions / min * 1 min / 60 sec * 2 * pi radians / 1 revolution gives 10 * pi or 31.4 radians per second.

Ex angular frequency of 300 Hertz:

300 Cycles / second * 2 * pi radians / 1 cycle = 600 * pi radians per second.

Hope this helps

Answer 2

Your question kind of jumps around and isn't very clear, but I'll give it a shot anyway.

In that formula you have for rotational energy, which I presume is a two dimensional problem, omega is just the angular speed in radians per second (or d_theta/d_t). Even though angular speed is the value used in the formula, a rotation is actually described as an angular velocity vector because there is a sense of revolution (clockwise or counterclockwise), but you don't care about that since you are solving for energy of rotation. Since you don't care about sense of rotation, you don't care in this case whether omega is angular velocity or angular speed. If it will clear up the confusion any, the formula would be better written as E=1/2 I |omega|^2 where the length of the angular velocity vector is the angular speed. It's no different than translational energy where E=1/2 m v^2. You don't actually use the direction of the velocity vector to calculate E in a translation. You just use the length of the vector v.

If you are referring to the angular velocity formula in three dimensions that is confusing you, the formula has the same units as the simpler expression for two dimensional case. It's just expressed in vector algebra. In three dimensions, the angular velocity has to define not only the rotational speed, but it has to define the sense of rotation and the plane about which the rotation is occuring as well. This is done by taking the cross product of r and v and dividing by magnitude of r squared. The cross product will define a vector in space. The direction that it points defines a plane normal to it for which the object rotates. The sign on the vector denotes whether the rotation is clockwise or counterclockwise in that plane. The length of the vector denotes the speed at which the object is rotating. Even in three dimensions, the rotational energy formula is the same as what you have and still uses the magnitude of angular speed omega to figure out the energy. It's just that in order to figure out what the angular speed is, you need to do the whole cross product computation to figure out the components of the resulting vector in order to calculate it's length. If you haven't been taught to do vector cross products, then your formula probably just wants you to use angular speed for a two dimensional problem.

Answer 3

(a.) d?/dt = angular velocity = 2t^3 - 3t^2 angular velocity at 2 seconds = 2(8) - 3(4) = 4 rad/s (b.) d²?/dt² = angular acceleration = 6t^2- 6t angular acceleration at t = 3 seconds: 6(9) - 18 =36 rad/s^2 (c.) 6t^2 - 6t = 0 t(6t - 6) = 0 t = a million sec.