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Dimensions question

  1. Apr 5, 2006 #1
    Hi guys,

    Hi guys,

    We are working with dimensions of equations in engineering, I do not completely understand the dimensions of sin(wt) and sin(w) and sin(t), are these all dimension less?

    w=angular velocity therefore dimensions of 1/T.
    t=time therefore dimensions of T
  2. jcsd
  3. Apr 5, 2006 #2


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    The amplitude of the periodic motion will establish the units. Most of the time, the units will be a displacement. You don't actually show the implied amplitude of 1.
  4. Apr 5, 2006 #3


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    As far as I know you can't take the sin of a dimension so sin(a) will always be dimensionless and can only be calculated if a is dimensionless as well. (This is just common sense telling me this so if anyone has any information to the contrary correct me)

    Like FredGarvin said it is the coefficient of the sin function that will determine the units.
  5. Apr 5, 2006 #4


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    Not exactly. w has units of radians per second. t has dimensions of seconds. I think that radians are generally considered to be dimensionless, or call it units of [1].

    It's not a sufficient question to ask what units sin(wt) has. It depends on the quantity that you are using that function to represent. For example,

    Air pressure -- P = P1 * sin(wt) could have units of N/m^2

    Voltage -- V = V1 * sin(wt) would have units of Volts

    The argument wt to the sin function is in radians in this form. Make sense?
  6. Jul 4, 2010 #5
    from Aditya Sonkar
    it's not dimension less
  7. Jul 5, 2010 #6
    w is for lower case omega, the angular velocity, in radians per second.

    When working with your homework problems, you will often be working with revolutions per minute. Showing how to convert to radians might be illustrative here.

    One revolution is 2*pi radians.

    Radians are said to be dimensionless because the trig functions are defined as having radians as their argument. There is no conversion factor once you are in radians, it is what I would call a "pure measurement". I would be interested in what a real mathematician would call it.

    rev = 2*pi radians
    rpm = rev / min
    = rev / [(60 sec / min) * min]
    = rev / (60 sec)
    = 2*pi radians / (60 sec)
    rpm= 2*pi radians / (60 sec)
    rpm= pi radians / (30 sec)
    Which lets us create a conversion factor:
    rpm/[ pi radians / (30 sec)] = 1
    30*pi*rpm/sec = 1
    The radians have been dropped ("dimensionless") because the definition of the trig function it will be plugged into is defined in terms of radians. You could keep writing it as part of the conversion factor, but most people don't.

    So if you are given 33 rpm and you want the angular velocity you use the conversion factor:

    w = [30*pi*rpm/sec] / (33 rpm) = [(30*pi)/33] 1/sec
    so we can directly find the angular velocity from the rpm.

    If you plug just w into the sin function, it gives an error because the argument should be in radians but you're giving it a value in inverse seconds. But multiply w by the variable t=time and the function generates the familiar sinusoidal shape.

    So you can either keep writing 'radians' in the conversion factor, and then you plainly see that w is in radians/second and w*t is radians. Or you can quit writing it out and just realize that you're treating the radians value as a pure number, and that your function is defined to interpret that pure number as a value measured in radian.

    One revolution is also 360 degrees, so if you can convert from degrees to radians by
    2*pi radians = 360 degrees
    duh, you knew that.

  8. Jul 5, 2010 #7
    The input of any transcendental mathematical function must be a dimensionless number and so is its output. That being said, out of [itex]\omega t[/itex], [itex]\omega[/itex] and [itex]t[/itex], which one is dimensionless?
  9. Jul 6, 2010 #8

    jack action

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    Sine, cosine, etc. are functions with one argument that must be an angle. Angles are considered dimensionless, but from my experience, it is a good idea to always keep them when deriving functions, as it is sometimes more helpful to understand what you are looking at. For example, it can be helpful to see the SI unit of torque as N.m/rad instead of just N.m (which is the unit for energy), such that:

    Torque = Energy / angle displaced [ = N.m / rad ]
    Power = Torque * angular velocity [ = N.m/rad * rad/s = N.m/s ]

    So for a internal combustion engine, the power is the energy per angle displaced times the angular velocity. The energy can be calculated from the fuel burned, the angular velocity is the rpm, but the angle displaced will be 1 revolution (2PI rad) for a 2-stroke and 2 revolutions (4PI rad) for a 4-stroke.

    The output of the trigonometric functions are purely dimensionless as they represents the ratio of one length over another:

    sine = opposite / hypotenuse
    cosine = adjacent / hypotenuse
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