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The first two images are from the resource from the Perimeter Institute, the
third was taken by Rolly Meisel
This lesson lets you take traditional physics and make it exciting and up-to-date. The first line of evidence for Dark Matter came from Newton's Universal Law of Gravitation, his second law of motion and the equations of circular motion. All you need to do this is get a copy of a great* free resource from the Perimeter Institute of Theroetical Physics. It consists of a 25 minute video and a variety of worksheets and activities that compliment the video. The video is divided into chapters so you can stop it in several natural places. To get your copy just go to The Mystery of Dark Matter and order a free teacher's kit.
A simple way to introduce the topic is to take a standard centripetal motion apparatus - tube, string, rubber stopper and washers - and modify it as described in the teacher's guide. Attach a 'turtle light' to the orbitting rubber stopper and place the washers in a plastic film canister. Do this to another one but put much less mass in the canister. Now, when you turn out the lights, the students will see what astronomers see - stars orbitting at different speeds, which provide evidence for central masses of different sizes.
Another way to make the role of the central mass more concrete and involving is to get a large (at least 1.5 m by 1.5 m) piece of really stretchy fabric as shown above and use it to create a gravity well. To work well, you need to have six students hold it level and really taut and another student needs to be underneath pulling on it really hard to make a good curvature.It helps if you provide a good handhold in the centre - I attached a small ball with rubber bands. Let your students experiment with launching speed and angle, eventually they will realize that it needs to be slow and perpendicular to the slope. To help focus the exploration, here is a worksheet for the students with answer sheet for the teachers.
Orbital motion yields the mass inside the orbit but appears to ignore the mass outside. This can be a tricky one to explain. To help your students understand here is a worksheet and answers for the teacher.
Finally, there can be considerable confusion between dark matter and black holes. The similarities and differences are explored in this worksheet and answers.
* In the interests of full disclosure, I must inform you that I may be biased in my assessment. I worked on this resource.