CALC A hollow, thin-walled sphere of mass 12.0 kg and diameter 48.0 cm is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by θ(t) = At^2 + Bt^4, where A has numerical value 1.50 and B has numerical value 1.10. (a) What are the units of the constants A and B?

Video transcript

Hey everyone in this video, we have a problem with a floor scrubber and the scrubber rotates about an axle attached to its center, an angle is given by theta T equals A. T squared plus B. T. To the four. Okay. We're told that data is in radiance, T is in seconds. That we're given values for A. And B. And were asked to determine the units of A. And B. Okay. All right. So if we're determining the units, we want to do a dimensional analysis. So let's start with this equation. We're given data. T equals A. T squared plus B. T. The four. Okay. And we want to know the units or find the dimension of these things. Okay. So we're gonna use square brackets to represent the dimension or the units. Okay, When we have quantities multiplied together, the units multiply together so we can have the unit of A separated from the unit of T squared and similarly here the unit of B Multiplied by the unit of T. to the four. Alright, so let's fill the information that we know. We're told that data is in radiant. Okay, so this value on the left is going to be radiant. Now on the right, we don't know the dimension of A or the unit of a. Okay, that's what we're looking for. So we're looking for this dimension or unit of A in the dimension or unit of B. Now T. We're told is in seconds. So T squared is going to be second squared, then we have the units of B again what we're looking for and then t to the four again, T in seconds. So we have seconds to the fore. Alright, now let's consider when we have two terms added together, when we're adding terms we need the units to be the same and then the unit of the sum will be the same of each part. Okay? So if we're adding a the unit of a. S squared plus a unit of B. S to the four, we actually need each of these to be equal to radiance in order for the sum to be equal to the radiance that we have on the left hand side. Okay, so we need radiant to equal the dimension of a. Times S squared. Okay? And we also need radiant. He goes to the dimension of B. S to the four. Okay, so we have this then we're gonna have a radiant plus a radiant. When we add we're going to get a unit of radiant which is what we want for theta, that value on the left hand side. Okay? So isolating the dimension or the unit of a radiant per second squared? On the right hand side. Isolating for B or the dimension or unit of B? We get radiant per second to the fourth. So those are the units we were looking for a is going to be units of radiant per second squared. B is going to be units of radiant per second to the fourth. Okay, that's going to correspond with answer F. Alright. And this is a really great tool to use when you're doing problems to make sure that you have the units right, that your units are consistent and that you're plugging in the right values in the right place is Alright. I hope this video helped. Thanks everyone see you in the next one.