In Episode 24 we are headed to a carnival to hop on rides that will have your head spinning and your stomach doing flips.
In Episode 24 we are headed to a carnival to hop on rides that will have your head spinning and your stomach doing flips. (1:01) Each rotational quantity has an analogous linear quantity that we encountered at some point throughout the year. The name of the game in rotational kinematics is to quickly learn the names of each variable and the units used to measure it. (1:54) The rotational kinematic equations are introduced, and should seem familiar. (4:03) Finally, we determine how far the door of our spinning amusement ride has traveled as it speeds up to top speed. (6:11)
The Question of the Day asks: (7:17)
What does the Greek letter that looks like the fishy thing represent and what is its actual name?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on rotational kinematics.
Let’s Zoom out:
Unit 7 – Torque and Rotational Motion
Topic - 7.1
Big Idea – Force Interactions, Change
The smell of popcorn and cotton candy is in the air, the steady clammer of people talking and laughing, and the screams of amusement riders all combine in a familiar orchestra. Carnivals are fun for lots of reasons, but they are also great places to study physics happening right in front of your eyes.
Let’s Zoom in:
As a child, one of my favorite ride was the Gravitron. Maybe you’ve ridden it? If not, you have probably at least seen it, or heard about someone whose stomach was upset after riding it. Afterall, the ride speeds up from rest to 24 rpm (rotations per minute) in just 20 seconds. And if you wanted to know the exact location of the door by that point in time, you could use rotational kinematics. Maybe you remember the word kinematics from earlier in the year? We used kinematic equations to describe motion in 1 and 2 dimensions in Unit 1. Well, the good news is the equations are essentially the same. The bad news… they are all different variables and measurement units. Keep Calm and Physics On!
The 5 variables you will be expected to be knowledgeable of are: angular displacement, angular velocity both initial and final, angular acceleration, and time. Hey! There’s a familiar one! Angular displacement will tell how much the position of a point moving in a circular path has moved, will be measured using our new friend the radian and the variable is the Greek letter theta. Angular velocity is measured in radians per second and we use the capital Greek letter Omega (it kind of looks like a rounded “W”). Angular acceleration is measured in radians per second squared or radians per second per second, and we use the lowercase Greek letter alpha, which some of my students refer to as “fishy thing”. All good! Whatever works for ya! Finally, we obviously use “t” as time and measure it in seconds. We should also agree on a direction too. It doesn’t really matter which way you consider positive, as long as you’re consistent, but most like to choose counterclockwise as positive since that is the way the quadrants of an X-Y coordinate plane are labeled.
Now that we know the ground rules, grab your Physics 1 equation sheet, and look for the rotational motion equations containing alpha… fishy thing. There should be two of them, and they should look very familiar. In fact, they are exactly the same as two of the linear equations we used earlier just with the rotational variables swapped in their place. One equation states that the angular displacement is equal to one half the angular displacement multiplied by the time squared added to the product of the initial angular velocity and the time. The other is essentially the definition of angular acceleration, or the change in angular velocity divided by the time. I know it isn’t written that way, but trust me… it is a great way to think about it, and if rearranged a bit, it does read that.
Let’s get back to the stomach churning ride of the Gravitron. If it takes 20 seconds to speed up from 0 rad/s to 24 rpm, we have everything we need in order to determine the angular acceleration … FINE fishy thing! We do however need to get our units in agreement, because 24 rotations per minute just isn’t going to work for us. We need radians per second. Since a single rotation around a circle is two Pi radians we can multiply 24 by 2 pi to arrive at 48 Pi radians per minute. Obviously, a minute is 60 seconds so by division, we find the final angular velocity to be 0.8 Pi radians per second or about 2.5 rad/s. With our definition of angular acceleration equation we see that 2.5 - 0 rad/s over 20 seconds is an average acceleration of 0.126 rad/s/s. If we assume it rotated in the counterclockwise direction, then all values were positive.
This still wouldn’t answer our original question, how far has the door rotated throughout the speeding up portion of the ride? To do that we need the other kinematic equation. One half of the 0.1256 rad/s/s acceleration multiplied by the 20 seconds squared gives us 25.2 radians. Since the ride started from rest, the initial angular velocity was 0 rad/s and we don’t have to worry about adding the other term.
Not too bad right? When you put in consistent effort throughout each unit all year long, it really pays off.
To Recap…
The linear kinematic quantities from Unit 1 have rotational analogies of angular displacement, velocity, and acceleration. Paired with time, these quantities can be utilized with the rotational kinematic equations to describe the motion of rotating objects.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at the moment of inertia and torque.
Today’s Question of the Day focuses on rotational kinematics.
Question:
What does the Greek letter that looks like the fishy thing represent and what is its actual name?