Episode 26 focuses on the energy associated with rotating objects. It's important to distinguish between translational motion and rotational motion.
Episode 26 focuses on the energy associated with rotating objects. It's important to distinguish between translational motion and rotational motion (1:06). An object that is rotating faster will not be able to be translating as quickly. (2:27) You will need to know the moment of inertia for each object in order to calculate velocity (4:30).
The Question of the Day asks: (7:10)
Assuming they are the same mass and radius, which wheel should you be able to get to rotate with more revolutions per second, a stationary exercise bike or a traditional bike as you ride?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on the energy associated with rotating objects.
Let’s Zoom out:
Unit 7 – Torque and Rotational Motion
Topic - 7.2
Big Idea – Force Interactions, Change, Conservation
A block of ice, a cylinder, and a hoop are released from rest at the top of an inclined plane. Assuming they are all of equal mass, the ice-incline surface is frictionless, and the hoop and cylinder are of equal radii, which one makes it to the bottom first? Until now, you would have assumed all of them would reach the ground at the same time. Not so!
Let’s Zoom in:
Before we begin, it is important to note that most of the motion you have studied is called translational motion. Motion in which an object’s axis of rotation, if it even had one, has translated or moved from its starting location in some way. Now however, we must also be concerned with the fact that some objects also rotate. For example a wheel on a level road is both translating as the car moves and rotating about the axle.
Until now, you would have looked at the scenario with the 3 objects on the incline and figured that all of these objects can be treated essentially the same. They change their heights by the same amount and gravity accelerates all objects at the same rate. Many, not all of you, would have decided a great way to solve this problem is to equate the gravitational potential energy of each object at the top of the incline to their kinetic energy gained by the time they reach the bottom. Solving for velocity, you would have noticed that their masses cancel and the velocities will all be the same. Since they all started at rest and traveled the same distance you would have found that the time to the bottom for each is equal.
And… for the ice block you would have been correct since it doesn’t rotate. BUT… as it turns out in the case of the cylinder and the hoop, it takes energy to rotate the hoop and cylinder about their rotational axes which are located in the center of their circular sides. This is known as rotational kinetic energy, and it is measured using units of Joules just like the other forms of energy. Translational kinetic energy is found by finding the product of ½ the mass of the object and the velocity squared. I am sure you can guess that the rotational kinetic energy is found by product of ½ the moment of inertia and the angular velocity squared. So, the gravitational potential energy each object-earth system possessed needs to be split up between the translational and rotational kinetic energies by the time they reach the bottom of the incline. Essentially, the object that is rotating faster will not be able to be translating as quickly.
As you may have guessed by now, the ice block will have the fastest translational velocity which is equal to the square root of the quantity 2 times little “g” times the vertical height of the incline. In order to determine the velocity for the cylinder and hoop objects, you would need to know their moment of inertia equations. The moment of inertia of a cylinder is found by ½ the mass times the radius squared. For the hoop, it is just mass times the radius squared. Since the moment of inertia of the cylinder is half of the hoop’s, it will have half the rotational kinetic energy, and therefore more translational kinetic energy. Let’s solve for the velocities at the bottom of the incline to prove it. Before you start to plug values into the rotational kinetic energy equation, it helps to convert the angular velocity to translational velocity, omega can be replaced with v over r. Combining like terms you will find that the velocity for the cylinder is equal to the square root of the quantity 4/3 times little “g” times the ramp height. For the hoop, the velocity is simply the square root of the product of little “g” and the ramp height. Not too shabby!
You should also feel comfortable now to determine the speed at the top tip of a meter stick that is standing vertically, and allow it to fall over without the bottom end slipping at all on the surface. Set the gravitational potential energy of the center of mass of the stick equal to the rotational kinetic energy of the meter stick just before it hits the ground. You would however need to know that the moment of inertia for the meter stick rotating about its end is ⅓ the mass times the length of the stick squared. You should be able to therefore arrive at a velocity equal to square root of 3 times little “g”, or 5.42 m/s.
To Recap…
In addition to the gravitational potential energy, elastic potential energy, and translational kinetic energy, objects can have rotational kinetic energy too. An object’s total mechanical energy can be comprised of any combination of these varieties. If the axis of rotation moves, then an object has translational energy, if the object rotates about the axis, then it has rotational kinetic energy.
Coming up next on the APsolute RecAP Physics 1 Edition, we look at the angular momentum of rotating object and its conservation.
Today’s Question of the Day focuses on rotational kinetic energy.
Question:
Assuming they are the same mass and radius, which wheel should you be able to get to rotate with more revolutions per second, a stationary exercise bike or a traditional bike as you ride?