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AakashPandita
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i solved a problem in all SI units and got certain value of angular acceleration. What is the unit of the value? Is it in radian/s^2 or what?
AakashPandita said:But in the answer it was only 25/s^2
Pythagorean said:That's not true; radians are physical units (same as degrees) and when you non-dimensionalize a system of equations, you have to account for that.
dauto said:That's not true. Look for instance at the equation that relates angular speed with tangential speed for an object in uniform circular motion: v=ωr, where v is measured in - say - m/s, w is measured in rad/s, and r is measured in m. Compare the units and you will see that radians are in fact dimensionless. The word radian is a reminder of what particular definition for an angle is being used, but it is not a physical unit.
What do you mean a convention to ignore the radians? The radians show up in one side of the equation but not the other. That mismatch of units is a no-no for dimensional physical quantities. How do you explain that except for the fact that angles are dimensionless quantities?Pythagorean said:It's a matter of convention that physicists ignore the radian in that case. Angular velocity technically has units of rad/s though. It is especially apparent when nondimensionalizing a system like the pendulum, where you treat the angle like any other dimensional quantity (breaking it into its dimensional constant and a non-dimensional variable).
That's right. Steradians are also dimensionlessIt becomes even more important when talking about steradians in three-dimensional systems. If radians weren't a physical unit, then neither would be steradians,
How so? I don't see it.and now you're implying that volume has the same dimensions as area.
dauto said:The radians show up in one side of the equation but not the other.
All fine and dandy except that speed is actually measure in m/s. You can't simply drop a dimensional unit. That would be wrong. But we do drop rads from the units all the time. How come? Hint: That's because they are not dimensional units.Pythagorean said:They're on both sides, we just ignore them in two places. Remember that an arc is actually (r)(theta). The tangential velocity, then, would be (rad)(m/s). So you have
V = wr
rad*(m/s) = rad*(m/s)
This all dropped in the canonical physics discussions of tangential velocity though, since we a priori describe it in the context of a circle (r being the radius, v describing motion around the perimeter of the circle).
Also, I guess we have to be careful about what we mean when we say dimension vs. physical unit. They are used interchangeably sometimes. All I'm challenging is your assertion that "radian isn't a physical unit". Not the rigorous definition of dimension because radians indeed have no "covering properties", only a directionality. But direction is important in physics (otherwise we wouldn't need vectors)!
Philip Wood said:I used to think that a radian wasn't a proper unit because an angle in radians is one length (arc length) divided by another (radius). But then it struck me: how do we measure mass? We might collide the body of unknown mass explosively with a body of known mass 1 kg (both at rest initially), and find the inverse ratio of the velocities after the bodies have separated. This will give us the unknown body's mass in kg. Is this process so very different? [A genuine question]
dauto said:All fine and dandy except that speed is actually measure in m/s. You can't simply drop a dimensional unit. That would be wrong. But we do drop rads from the units all the time. How come? Hint: That's because they are not dimensional units.
atyy said:dauto is of ciurse right. You can treat 8 m as 8 times a standard metre. In a ratio of lengths metre/metre is 1. If a quantity is multiplied by 1, we can always leave out the multiplication by 1.
dauto said:In this example you're not measuring the mass. You're actually measuring the ratio between two masses which is indeed adimensional.
Philip Wood said:How else do you measure mass? Aren't you - ultimately - measuring a ratio?
Pythagorean said:I don't disagree with that but it's an inappropriate analogy. If you try to equate (1/s) to (rad/s), you're saying that 2*pi = 1.
Philip Wood said:How else do you measure mass? Aren't you - ultimately - measuring a ratio?
AlephZero said:Regardless of the learned discussions about ratios, if somebody tells me "the angle is 42", without any context, my first question would be "is that 42 radians, or 42 degrees, or 42 grads, or 42 complete revolutions?"
They are all "dimensonless", but not all the same size.
But then I'm an engineer not a physicist - but I remember we once hired a physicist to do some computer programming, and he couldn't understand why engineers wanted to input angles in degrees measured clockwise from the vertical, instead of in radians measured anticlockwise from the horizontal
Pythagorean said:If the NIST standard (see: The International System of Units: Physical Constants and Conversion Factors) for rad/s as a physical unit isn't enough, there's a more rigorous discourse here:
http://khimiya.org/volume14/radian_.pdf
But there's really nothing new there on top of what I've already said. It also highlights the important difference between "dimension" and "physical unit". If there's any confusion about the physical quantity being measured... it's called angle.
atyy said:http://physics.nist.gov/cuu/Units/units.html
So looks like Pythagorean has a point. Radians is a way of multiplying by 1 (see Table 3). Indicating multiplication by 1 is optional but can aid in clarity (see Table 4).
dauto said:If that were true than we would all have been very sloppy every time that we wrote the expression sin(θ) in our lives.
dauto said:True, angles are physical things but they happen to be adimensional physical things.
srijag said:The unit is indeed dimensionless. Radian is dimensionless. If you try to evaluate the dimensions of angular acceleration, it is T^-2. So the units of angular acceleration being s^-2 makes sense.
The unit of angular acceleration in SI units is radians per second squared (rad/s2). This is equivalent to the unit of acceleration in linear motion, meters per second squared (m/s2), but instead measures the change in angular velocity over time.
Angular acceleration and linear acceleration are related through the radius of rotation. Angular acceleration is equal to linear acceleration divided by the radius of rotation. This relationship can be seen in the equation α = a/r, where α is angular acceleration, a is linear acceleration, and r is the radius of rotation.
Yes, angular acceleration can be negative. A negative angular acceleration indicates that the object is slowing down or decreasing its angular velocity. This can happen when an object is decelerating or when it is rotating in the opposite direction.
Angular acceleration and angular velocity are both measures of an object's rotation, but they are different quantities. Angular velocity measures the rate of change of angular displacement over time, while angular acceleration measures the rate of change of angular velocity over time.
Angular acceleration can be measured using a device called an accelerometer, which measures the rate of change of angular velocity. It can also be calculated using the equation α = Δω/Δt, where α is angular acceleration, Δω is the change in angular velocity, and Δt is the change in time.