(P.15) Angular Measure expressed in radians

[gcombina]

Which equation is valid only when the angular measure is expressed in radians?
a) α = Δθ / Δt

b) ω= Δω / Δt

c) ω^2 = ωo^2 + 2αθ

d) ω = Vt/r (here T is a subscript)

e) θ = 1/2αt^2 + ωαt

Answer is D but why??

* I am totally lost so I can not show work

 

[Nathanael]

So equation D is saying that angular velocity multiplied by the radius is equal to tangent velocity.

Think of an object on a string spinning around in a circle. The length of the string is r (which is the radius of the path of the object). The angular velocity is $\frac{Δθ}{Δt}$ but the tangent velocity is $\frac{length.of.arc .subtended.by.Δθ}{Δt}$

Multiplying an angle by the radius will give you the length of the arc subtended by that angle ONLY if the angle is measured in radians (that's the definition of the radian, and the reason it's such a useful unit)
That is why that statement is only true if it is measured in radiansHope this was clear

 

[Nathanael]

a) α = Δθ / Δt

b) ω= Δω / Δt

Also shouldn't it be:?

a) ω = Δθ / Δt

b) α = Δω / Δt
P.S.
The reason the other equations don't need to be measured in radians is that they just involve angles, so it doesn't matter if it's radians, degrees, or something else, because the answer will come out in the same angular units.

Equation D is the only one that deals with angles AND (linear) distances.

 

[gcombina]

"The reason the other equations don't need to be measured in radians is that they just involve angles, so it doesn't matter if it's radians, degrees, or something else, because the answer will come out in the same angular units."
(so if something is measured in degrees, the answer will be defaulted into radians?)

Equation D is the only one that deals with angles AND (linear) distances". (I think when something is measured in angles then we have degrees right? so if you say that equation D only deals with angles and linear distances, isn't angles always measured in degrees?)



PS:
Equation D shows angular velocity which is by default measured in radians right? basically that is the hint that I got but still a little confused with these equations.

 

[haruspex]

"The reason the other equations don't need to be measured in radians is that they just involve angles, so it doesn't matter if it's radians, degrees, or something else, because the answer will come out in the same angular units."

(so if something is measured in degrees, the answer will be defaulted into radians?)

No. Nathanael is saying that in an equation like ω = dθ/dt it doesn't matter whether the angles are measured in radians or degrees as long as you are consistent. It's exactly the same with linear distances; the equation v = ds/dt is valid in various units as long as they match: m/s, m and s, or mph, miles and hours, etc. If you were to change units it would multiply both side of the equation by the same factor.

Equation D is the only one that deals with angles AND (linear) distances".

(I think when something is measured in angles then we have degrees right?)

No, you can measure angles in degrees or radians, or revolutions, or whatever you care to define.
I take a slightly different view from Nathanael's. To me, what's special about equation D is that on the RHS you have a velocity divided by a distance which is not in the same direction. Indeed, it's at right angles.
Suppose I'm going in a circle radius r at a steady speed v. After time t, I've gone a distance vt. If I divide that by the radius of the circle I get a measure of how far round I've gone, so vt/r is giving that angle in some units. When I've gone distance 2πr I will have gone once around the circle, so the units must be such that an angle of 2π in those units is 360 degrees. This serves to define the unit radian.

 

[dauto]

Also note that it is important to learn to use subscripts. V_t is the tangential velocity while Vt is just the velocity times time. These are two different things. Use the go advanced button and look for the subscript button that looks like a X₂.