# Inclined Plane

 Physics: Geometric Shapes, Center of Mass Grade Range: Elementary School, Middle School, High School Format: Hands-on, Stage

This demonstration is easy to set up, and it will amaze students that the objects travel down the slope at different speeds. This demonstration can be explained in different ways for different age groups.

## Materials

• Inclined Plane
• Inclined Plane Box:
• Plastic and Metal Spheres
• Plastic and Metal Large Hollow Cylinders
• Plastic and Metal Medium Filled Cylinders
• Small Aluminum Hollow Cylinder (Thin Walled)
• Small Brass Hollow Cylinder (Thin Walled)
• Small Brass and Aluminum Filled Cylinders
• Small Aluminum Hollow Cylinder (Thick Walled)
• Inclined Plane Catch Box
• Inclined Plane Wedge

## Safety Precautions

Please read the Physical Demonstration section of the Demonstration Safety page before performing this demonstration.

## Demonstration

Preparation
• Set up the inclined plane: Lay the inclined plane face-up across a table. Use the wedge to set the height of the inclined plane, and set the plane to an angle between 20 and 30 degrees. Set the catch box at the end of the inclined plane, and set out all the cylinders and spheres either in front of the plane or to the side.
Presentation (Stage)
1. Ask for a student volunteer, and have them hold the two spheres and tell you which one is heavier. Ask the audience if they think that the weight of the sphere will affect how quickly it will roll down the slope, and take a quick vote. Have the student hold both spheres at the top of the inclined plane, and instruct them to release both spheres at the same time when you say "GO". Give them a "Ready, set, GO", and watch the spheres go. They will roll at the same speed!
2. Repeat step one with the large hollow cylinders. They will roll at the same speed!
3. Repeat step one with the medium filled cylinders. They will roll at the same speed!
4. Repeat step one with the small aluminum hollow cylinder (thick walled) and the small aluminum filled cylinder. They will roll at different speeds!
5. Insert the small aluminum filled cylinder into the hollow cylinder. Now give the student the small brass filled cylinder and repeat step one. They will roll at the same speed!
6. Put the sphere, a small hollow cylinder and a small filled cylinder on the inclined plane together and ask the audience to vote on which one will roll the fastest. Release all three at the same time. The sphere will finish first, followed by a filled cylinder, and last is the hollow cylinder!
Presentation (Hands-on)
1. Allow students to come up and race the different shapes. Have them race the matching metal and plastic shapes first, then have them race the different shapes to see which ones move the fastest.
2. For more curious students, have them try the following:
• Compare the speeds of the different hollow cylinders. Why don't the hollow cylinders all move at the same speed?
• Compare the speeds of the different filled cylinders. Why do the filled cylinders all move at the same speed?
3. Be sure to watch this demonstration, and don't allow students to throw or push the shapes down the slope.

## Why This Works

### Short Explanation

When something rolls down a hill or slope, we know that it will accelerate because of the force of gravity. However, we don't usually think about how much it will accelerate, which depends on two things: the shape that is rolling, and how hollow the shape is. When we rolled the solid spheres, we see that they all accelerated at the same rate, even though they were made of different materials. This is because they are the same shape, and so they will accelerate the same down a slope! We saw the same thing happen with the solid cylinders, but then we saw that a hollow cylinder rolls slower than a solid cylinder. This is because a hollow shape has a different center of mass than a filled shape. The Center of Mass of an object is a spot in space where the mass of the object is centered around. The center of mass doesn't have to be at a spot in the object's mass. For a hollow shape, the center of mass is in the hollow area, so it has to transfer any motion along the outside, which takes longer. The filled cylinder can transfer motion through its center, since the center of mass is within the object's mass. It is faster to go through the center than to go around the outside, so the solid cylinder will always roll faster! The solid cylinder, however, rolls slower than the solid sphere. This is because every spot on the outside of a sphere is the same distance from its center, while every spot on a cylinder is not the same distance from its center. A solid sphere will always roll faster down a slope than a solid cylinder!

### Full Explanation

The inclined plane is a common introductory topic in physics. On a plane that is at a certain angle you can place an object, and then calculate the forces acting on the object as well as the acceleration of the object. In this case, however, we are looking at rolling objects on an inclined plane. For rolling objects, the calculations we need to do are a little different.

We see that the two solid spheres roll at the same rate. Even though they were made of different materials, and therefore would have different amounts of force acting upon them, they roll down the slope at the same rate because they are the same shape. We also see that the solid cylinder rolls slower than a solid sphere, and a hollow cylinder rolls slower than a filled cylinder. When we are looking at rolling shapes, we have to look at the shape's Moment of Inertia, and how it corresponds to the shape's velocity:

 Shape Moment of Inertia Velocity at Bottom of Incline Solid Sphere I = 2/5 m*r 2 v = (10/7 g*h)1/2 Solid Cylinder I = 1/2 m*r 2 v = (4/3 g*h)1/2 Hollow Sphere I = 2/3 m*r 2 v = (5/6 g*h)1/2 Hollow Cylinder I = m*r2 v = (g*h)1/2

Where m is mass, r is radius, g is gravity, and h is height of the incline.

The moment of inertia relies on where the center of mass is for the shape, and therefore how the mass is distributed throughout the shape. If a shape has mass where the center of mass is located, such as the solid sphere or solid cylinder, then it will have a lower moment of inertia due to it. If a shape does not have mass where the center of mass is located, such as the hollow sphere or hollow cylinder, then it will have a higher moment of inertia. The higher the moment of inertia, the more energy is required to keep rotating, and therefore less energy is available for linear movement.

We see with the filled cylinders of varying size that the velocity is independent of the size. This is also seen with the equations we have above; Although the radius and mass of the object is used to calculate the inertia, the end velocity only uses a constant, the acceleration of gravity, and the height of the incline. This is because the end velocity is independent of the mass or the size of the object, and is dependent on the dimensions of the object itself, and therefore on the object's moment of inertia.