What makes an object easy or difficult to rotate? Rotational inertia is a quantity that describes an object’s tendency to maintain its rotational motion about an axis.
What makes an object easy or difficult to rotate? Rotational inertia is a quantity that describes an object’s tendency to maintain its rotational motion about an axis. (1:02) Torque is equal to moment of inertia multiplied by the angular acceleration (1:44). Try opening a door by pushing it very close to the hinges. Feels silly right? (5:40) The episode ends by putting the brakes on a bicycle wheel (6:53).
The Question of the Day asks: (8:11)
If the radius and mass of a ball are both doubled, how does the moment of inertia change?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on a quantity called the moment of inertia and torque.
Let’s Zoom out:
Unit 7 – Torque and Rotational Motion
Topic - 7.1
Big Idea – Force Interactions, Change
Whether you are opening a door, cranking a wrench, or braking as you approach a red light, you have the concept of torque to thank. The cause for objects to change their linear velocity is force, the cause for objects to change their rotational velocity is torque, but not all objects rotate equally.
Let’s Zoom in:
What makes an object easy or difficult to rotate? I mean, not all objects rotate with the same amount of ease as a bike wheel for example. If you try to rotate the Earth, it would be way more difficult than rotating a bicycle wheel. Why is that? You may have guessed that the more massive the object is, the more difficult it is to rotate. While mass is a factor, it isn’t the ONLY factor when determining moment of inertia. It turns out the larger the radius is of a rotating object, the harder it is to rotate as well.
Inertia is the tendency of an object to maintain its motion. Rotational inertia, designated with a capital “I,” is a quantity that describes an object’s tendency to maintain its rotational motion about an axis. When objects were only moving in straight lines, the inertia was dependent upon mass. You can think of the moment of inertia sort of like rotational mass. Your teacher will probably provide you with various formulas for the moment of inertia for different shapes. Common objects are points, disks, rods, and spheres. On the Physics 1 exam, you will always be provided with these equations if it is necessary or you will just be given the moment of inertia. The units for moment of inertia are kg*m^2. and the equation will always be the product of mass, the radius squared, and some numerical multiplier. For example, the moment of inertia for a disk rotating around the center of its circular face is ½ m*r^2 while a solid sphere’s moment of inertia is ⅖ m*r^2.
Ok, so even if we wanted to get a bicycle wheel to begin rotating and we know its mass, radius and the equation to determine its moment of inertia, how quickly it speeds up or slows down must depend on how hard you spin it. That is torque. We used Newton’s 2nd Law to relate the mass of an object to how quickly it accelerated, and with rotational motion we can relate the moment of inertia to the angular acceleration of an object… yes… fishy thing is back. Newton’s 2nd Law for Rotation explains that the net torque applied to an object is proportional to its moment of inertia, but inversely proportional to its angular acceleration. Essentially, the same equation we saw before with the 2nd Law, but now Torque is equal to moment of inertia multiplied by the angular acceleration. The units for torque are therefore kilogram meters squared per second squared or Newton meters.
All of that can be a bit mathematical and somewhat conceptually confusing. Fortunately, there is another way to determine torque, and it makes the whole concept make much more sense. Torque can be found by multiplying the applied force by the distance between the axis of rotation and the point of the applied force multiplied by sine of the angle between the force and the radius of rotation. I know that sounds complicated, but many problems have the angle set to 90 degrees so that sine 90 equals 1. In practical terms, the handle on any door is placed at nearly a maximum distance from the axis of rotation, the hinges, so that you have maximum torque when you try and open it. Decrease that distance, and the door becomes A LOT harder to open. Don’t believe me? Try opening a door by pushing it very close to the hinges. Feels silly right? Another ill informed physics person might try and apply force along the door towards the hinges so that the angle between their force on the door and the door itself is 0 degrees. Sine zero is 0, so you will get no rotation out of that door since you will not be applying any torque. Finally, the weaker you push on a door, the less torque you have. All of these factors combine to allow you to more easily open doors, but also tool manufacturers make a lot of effort to maximize your torque as well. Think about most tools, long handle, check. Ability to apply force at 90 degrees to the lever arm, check. Look around your parent’s workbench. My guess is you will find tons of examples of torque maximization at work.
Back to the bicycle wheel. When you apply the brakes to most bikes, a set of rubber pads clamp down and use friction to slow the bike. If your 0.5 kg wheel has a radius of 50 cm, or 0.5 m and you were cruising along with an angular velocity of 6*pi radians when you apply your brakes, and it takes you half a second to come to rest, thennnnnnnnnn….you can find the torque of the brakes by multiplying the moment of inertia for the wheel by the angular acceleration. The moment of inertia for a wheel is m*r^2 and the angular acceleration can be found by dividing the velocity change by the time. Little bit of math...and the torque of the brakes is -4.71 N*m.
To Recap…
Moment of Inertia is like rotational mass, it tells how hard something will be to get it to rotate. Torque is the rotational analogy for force, and can be found by the product of force, sine theta, and radius OR by multiplying the moment of inertia by the angular acceleration.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at the energy of rotational motion.
Today’s Question of the Day focuses on rotational kinematics.
Question:
If the radius and mass of a ball are both doubled, how does the moment of inertia change?