Conservation of Linear and Angular Momentum During a Collision

Students will:

• review the concept of conservation of linear momentum by calculating the linear momentum of a system before and after a collision.
• calculate the angular momentum of a rotating object.
• recognize that the angular momentum after a collision does not arise from the linear momentum before the collision, that is, that linear momentum cannot be converted to angular momentum.
• recognize that a particle moving in a straight line can have angular momentum with respect to a point in space.
• determine the factors that affect the angular momentum of a particle moving in a straight line.
• use measurements and calculations to show that for this example both linear and angular momentum are conserved to within a few percent.

This activity is intended for use in any introductory physics course where students are expected to understand angular momentum of a particle. This is not usually covered in algebra-based introductory courses, but it is covered in many calculus-based courses, and in AP Physics C Mechanics. The activity is intended to allow students to discover the idea of the angular momentum of a particle by building on prior knowledge gained thus far in the course. Specifically, students should already know:

• how to use frame-counting and position measurements to find the velocity of objects in a direct measurement video.
• how to calculate the linear momentum of an object and a system of objects.
• how to calculate the rotational inertia and angular momentum of a rotating object, such as a rotating block of wood.

The goal is for students to use what they already know to realize that the angular momentum of the block after the collision cannot be caused by converting some of the linear momentum of they marble to angular momentum, because linear momentum is conserved during the collision. This gives rise to the question: where did the angular momentum come from? Students then generate their own list of the factors about the motion of the marble that contribute to the angular momentum of the block.

Students can work alone or in small groups. The instructor can lead discussion at intervals during the activity to help make sure students are proceeding along a productive track.

The activity can be completed in one class period of about 50 minutes if students are focused and have experience making measurements from this type of video.

Students will need to download the QuickTime version of the video to their computer. Viewing the video in QuickTime allows students to control the video and use their keyboard to move one frame at a time. This is essential to using direct measurement videos. Students may need to be reminded that viewing the video within the web browser will not allow them to advance the video frame-by-frame, which makes measuring time intervals very awkward. In most browsers, use ctrl-click, right-click, or two finger click and save the file to the computer. Other than the QuickTime player, students do not need any additional software to analyze the video.

Ideally, students use computers to view the video and make measurements on their own or in small groups. Alternatively, the video can be projected and the class can make measurements together. Once the measurements are made, students can perform calculations and answer questions to complete the activity on their own or in small groups.

Download the QuickTime file: Marble collides with wood block (Quicktime Video PRIVATE FILE 7.8MB Feb8 13) here.

The document here shows one set of instructions that could be used with this activity. However, these are very descriptive instructions that do not allow room for student exploration. An alternative approach would be to give the students the video and a simpler set of questions. For example, students could be asked to determine whether linear and angular momentum are conserved. For this approach, students would need experience with making measurements from videos.

Student Instructions for Conservation of Linear and Angular Momentum Activity google doc

For students to reach useful conclusions from the video, they must make accurate measurements from the video. For example, if they measure the initial velocity of the marble incorrectly, the conservation of momentum calculations will be wrong as well, and they activity will not be effective. To prevent this, pause the activity and discuss the measurements as a class before students progress to the calculations. Here are the key measurements:

1. frames for marble to travel 27 cm rightward before collision: 6 frames
2. frames for marble to travel 27 cm leftward after the collision: 60 frames
3. frames for the center of mass of the block to move 27 cm to the right: 79 frames
4. frames for the block to rotate one revolution, end-over-end: 79 frames

For this analysis, the angular momentum around the long axis of the rectangle is low enough that it can be ignored. However, the time for one half rotation on the long axis is 67 frames.

When students have completed the section called "Questions Part One", the class discussion to lead students to the idea that the magnitude of the angular momentum of a particle around a point in space is:

l=rmv

Equipped with this, students can complete the analysis. Students should conclude that both linear and angular momentum are conserved to within a few percent.

Here is a google spreadsheet showing the measurements and calculations for this video

The instructor can determine how much guidance to provide. If the classroom culture is one where students are comfortable with inquiry-based work, the instructor can provide little guidance. The materials presented here provide maximum guidance, and are offered to give the instructor the option to determine how the activity is conducted.

In most calculus-based introductory courses, students would be expected to be able to solve a written problem with a situation like the one in the video. For example, question 3 on the 2005 AP Physics Mechanics C exam and question 2 on the 1998 exam feature similar situations. Either of these problems can be used as an assessment. This College Board list of questions from previous AP Physics exams has links to these problems, along with solutions and grading rubrics.

Alternatively, students could be assessed in writing or verbally. For example, students could be asked to identify a real-life situation where linear and angular momentum are both conserved, and to identify the quantities that determine the final linear and angular motion of the objects.

The hyperphysics website has a helpful page on angular momentum:

Here is a list of rotational inertia formulas for many objects, including a block (called a cuboid on this list)