Objects can move rotationally and translationally! So we need to find ways to describe both. Δ𝛳 is the change in the angular position, in radians, and is the rotational analog for displacement Δx ѡ is angular velocity in the units radians per second, and is the rotational analog for velocity(v). ɑ is angular acceleration in the units of radians per second squared and is the rotational analog for acceleration(a).
You may notice that there's no such thing as angular time here, which is great for us as its one of the ways we can tie these two worlds together with something other than radius!
Translational vs. Rotational:
Below we can see the comparisons between translational and rotational motion, and you may notice things are eerily similar.
The formulas that work for both, other than the last one, require the object to be rolling without slipping, which essentially means that there is a torque on the object. For example, look at the picture below:
1) Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of 0.250 rad/s2. After the wheels have made 200 revolutions (assume no slippage): (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train? ([Taken from Lumen Learning)]
Remember the equation between theta and radius: Δx=r𝛳
Suppose a yo-yo has a center shaft that has a 0.250 cm radius and that its string is being pulled.
(a) If the string is stationary and the yo-yo accelerates away from it at a rate of 1.50 m/s^2, what is the angular acceleration of the yo-yo?(b) What is the angular velocity after 0.750 s if it starts from rest?
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