Linear and Rotational Kinetic Energy (Gregory Hanson)

Title: Linear and Rotational Kinetic Energy!

Principle(s) Investigated: First Law of Thermodynamics, Linear kinetic energy, rotational kinetic energy.

Standards: HS-PS3-2. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative positions of particles (objects).

Materials:

• Smooth board (such as a plank of wood or even cardboard if it is solid enough)
• Books
• Stopwatch
• Scale
• Ruler or tape measure
• Solid spheres with different radii (marbles, wooden balls from craft stores, golf balls, bouncy balls)
• Disks with different radii (available at craft stores as well as hardware stores)
• Hoops with different radii (available around the house or at your nearest hardware store such as Lowes or Home Depot).

Procedure:

1. Gather a collection of two (2) spheres with different radii, two (2) disks with different radii, and two (2) hoops with different radii. Examples of items in image below.

1. Measure the mass and diameter of each item and record onto the data table. (remember which item is which!)
2. Place the smooth board on top of a book to create an incline in which the rise is 10% of the run (example: if your board is 1m long, the height of the incline would be 10cm).
3. Place your two spheres at the same level on the inclined board, release, and record the time it takes to reach the bottom in your data table.
4. Repeat step 4 with your disks and hoops.

Do your spheres accelerate at the same rate? What about your hoops? Your disks?

Student prior knowledge: Students should know the concepts of linear kinetic energy, gravitational potential energy, and conservation of energy.

Explanation: When an object is dropped, it only possess linear (or translational) kinetic energy. When you roll an object down an incline, however, it possesses two kind of kinetic energy: linear and rotational. If you look at a bike in motion, the wheels rotate as well as move linearly (translate). If you were to place the bike upside down and rotate the wheels, they would not move linearly, but would still rotate. They would possess rotational kinetic energy, but not linear kinetic energy. The wheels rotate about their axles, but the axles remain in the same location. In regards to the total energy of the system, as the bike is moving, the energy is divided between rotational kinetic energy and linear kinetic energy (because the bike's wheels are both rotating AND moving linearly).

In our system, there exists potential energy while the objects are at the top of the ramp. All of this potential energy is converted into kinetic energy as the objects roll down the ramp. The only factor that determines how much time it takes for the object to reach the bottom is translational kinetic energy. The fastest object will have the highest level of translational kinetic energy. The slowest will have the highest level of rotational kinetic energy.

Why do some objects reach the bottom much quicker than others? It does not solely depend on the mass of the object. It also depends on the distribution of mass around the axis of rotation of the object. If the mass is concentrated closer to the axis of rotation, the object will have higher linear kinetic energy. If the mass is further away from the axis of rotation, the object will have much more rotational kinetic energy. In comparing our objects, the spheres have mass that is concentrated closer to the rotational axis, the discs had mass that was uniformly spread, but less concentrated near the rotational axis, and the hoops had mass that is concentrated much further from the rotational axis.

The total energy of the system is given by the following equation: E_k= ½ mv² + ½ Iw²

(Figure 1. The moment of inertia equations for hoops, disks, and solid spheres)

From Figure 1 we can see that as the mass of an object becomes more concentrated away from the rotational axis, the inertia increases.

Questions & Answers:

1. Do you think a sphere will always beat a hoop down the ramp?

Not necessarily! It depends on the mass and the radius of each object. If a hoop has a much smaller radius and mass than a sphere, it could potentially beat the sphere down the ramp.

2. Why would choosing to place smaller-radius wheels on your car be a better option than larger-radius wheels?

The car's engine has the same energy output regardless of the wheel choice. If you choose a smaller-radius wheel for your car, you concentrate more of the wheel's mass toward the rotational axis, allowing your car to have more linear kinetic energy rather than more rotational kinetic energy. This can help save you money over time by using less gas.

Applications to Everyday Life: Racing- focusing on wheel design can allow for much quicker acceleration of the car even with the same engine output. This will allow for greater amounts of linear kinetic energy vs rotational kinetic energy. In space, rotational energy plays a role because of the lack of friction from air. If a space shuttle is acted upon by an outside force at the wrong angle, the space shuttle will have much more rotational kinetic energy than linear kinetic energy. This means that the shuttle will simply rotate instead of move where the astronauts want it to go. In regards to a satellite's orbit, this could spell disaster if the rocket boosters are fired incorrectly. Instead of correcting the orbit, the boosters would simply cause it to rotate, which would make it much more difficult to correct its position. In ice skating, the ice skaters spin on their toes. If they hold out their arms, they spin more slowly than when they bring their arms closer to their body. This represents the conservation of rotational kinetic energy because if the total energy remains constant, as you concentrate the mass more closely to the axis of rotation, the angular velocity increases, causing the figure skater to spin more quickly.

Videos:

(Note: The lab activity presented by this teacher is slightly different. He goes a bit more in-depth, which you can do if you would like! It seems geared more towards late high school students)