Conservation of Energy 

University of British Columbia

PHY420 -- 2017/2018

Abdullah Abdulqader

Introduction:

On which of the four paths shown in the following figure would a ball roll down from point A to point B and end up with the highest velocity at point B?

This webpage documents my attempt to answer the question above using a physics demonstration that I built.

This demonstration aims to improve students’ understanding of the following concepts:

This demonstration aims to improve students’ understanding of the following concepts:

• Conservation of Mechanical Energy (main concept):
  The total mechanical energy of a system is conserved as long as the forces acting on the system are conservative forces. That is, the sum of the potential energy and the kinetic energy of a system is constant.
  
  
• Theory vs. Practice:
  It is important that students understand that the physics problems they solve in their textbooks do not portray how the real world works.

Students are expected to have a good background on the following:

• Distance, Time, Velocity: Velocity is the distance traveled over a time period.
  
  
• Friction, Drag: Friction and drag slow moving objects.

After this demonstration, students are expected to be able to:

• Explain the principle of conservation of energy: The sum of the potential energy and the kinetic energy of a system is constant.
  
  
• Understand how friction and air drag affects the outcome of the demonstration: Friction from the tracks and from air slow down the ball as it slides down the tracks. The kinetic energy lost by friction is converted to heat.

Apparatus and Construction:

The following are the components of this physics demonstration, only the first component requires construction:

1. Four tracks:
   
   Figure 1 shows the rough shape of the tracks as viewed from the side. Point “A” is the starting point of the tracks. All tracks have the same height at points “A”, “B”, and “C”. The only difference between the tracks lies between points “A” and “B”. These tracks were made out of hardboard (pressed wood), each track will be 0.6 meters high and 1.2 meters long.
   
   Each track consists of two identical slabs of hardboard separated by 1.5 cm gap. Figure 2 shows a cross-sectional view of a track and a ball mounted on top of it. This design provides two features: that the friction between the ball and the track is minimized by reducing the contact area to exactly two points, and the ball will have a “trail” on the tracks that keep the ball moving in a straight line.
   
   
2. Marble Balls:
   
   I chose marble balls over any other rollable object because marble balls are usually made of glass or steel, which will create relatively small friction over balls made of other materials.
   
   
3. Two Timer Gates:
   
   The first timer gate will be placed at point “B” and the second will be placed at point “C” (Figure 1). These timer gates will be used to calculate the time it took the ball to travel the distance between them.
   
   
4. Ruler:
   
   To measure the distance between the timer gates. And to measure the height of the tracks.

I constructed the tracks using a laser cutter to get the curves to be as smooth and continuous as possible. But you can construct it any way you want. Here are some design details:

• The first track has 40-45 degrees slope line with a little curve before point “B” to have a smooth continuous transition between the slop line and the straight horizontal line. It is important that the slope angle is large enough such that the ball starts rolling with an initial momentum that will easily overcome friction forces.
  
  
• The second track will roughly look like a wide parabola, followed by the horizontal line.
  
  
• The third track will have a very steep curve as shown in Figure 1.
  
  
• The fourth track will be a bit different from the other tracks, part of it will be under the horizontal line as shown in Figure 1.
  
  
• These tracks do not need to perfectly represent mathematical equations (a parabola or a 6th-degree polynomial).The goal of having different track shapes is not to show how different curves affect the velocity of the ball, but to show that the ball’s velocity at point B is independent on the path it took to get there.
  
  
• A 2.0x1.0 m slab of wood will be used as the base on which all tracks will be mounted. Each track will take 1/4 of the base’s width, which is 0.15 meter. This space allocation may be reconsidered depending on how close the gate timers must be to the rolling ball.
  
  
• Each track will consist of two identical wood planes (Figure 2). These planes will be 0.5-1cm thin, a lot thinner than the diameter of the marble balls. Two 1.5 cm wide blocks of wood will separate the two planes of a tack and hold them parallel to each other.
  !
  
  
• The edge of all tracks is smoothed then tested. Although steel may cause less friction than smoothed wood, it may be difficult to make the tracks from steel, and the cost will be higher. Besides, how friction affects the expected theoretical outcome of the demonstration is part of the lecture, so using smoothed hardboard should be fine as long as the friction does not affect the outcome by more than 15-20% of the expected theoretical outcome. If the expected final velocity of the ball is 3.5 m/s, then the measured velocity should not be less than 3 m/s, or else the error may be too big to demonstrate the concept of energy conservation.

This is how one of my tracks looked like:

Pedagogy:

A quick demonstration of rolling the ball down the tracks will be done to show them how the set up works, then, in the first round of predictions, students will be asked the following question: On which track would the ball gain the largest velocity when it reaches the bottom of the track?

The goal here is not to correct the students’ answers, but to get them to gain their attention and excitement.

Students will be asked to suggest ways to measure the velocity of the ball. I will show them the tools I have: a ruler and timer gates. How the timer gates work will be explained to them.

Once the setup is ready to measure the velocity of the ball down the tracks, the demonstration will be repeated three times for each track and the average velocity of each track will be recorded. Of course, why the average is taken will be explained. 

Hopefully, if the friction of the tracks were not an issue, the average velocities should agree to some degree. Now, the real topic of the demonstration can be motivated: Why do the velocities agree?

In this part of the presentation, the principle of conservation of mechanical energy is introduced. First, students will be reminded of what energy, potential energy, and kinetic energy mean. Then, The principle of conservation of mechanical energy will be introduced along with the related equations. The goal here is to deliver the concept and show how it can be used to solve simple physics problems.

The mechanical energy E of a system is the sum of its potential energy U and the kinetic energy K. The mechanical energy is conserved as long as the forces acting on the system are conservative.

The potential energy of the ball at the top of the tracks is the same for all tracks because the height of the tracks is equal. The gained kinetic energy from all tracks is, theoretically, the same because the balls end up in the same height (setting the reference point to be the bottom of the tracks).

U_i + K_i = U_f + K_f

U_i + 0 = 0 + K_f

mgh_i = 0.5mv_f^2

v_f = √(2gh)

The theoretical outcome is then v_f=√(2*9.81m/s^2 *0.5m)=3.132m/s. This outcome will be compared to the measured velocities.

Why is the measured velocity less than the theoretical velocity? Students will be asked to brainstorm together and write down their answers in their worksheet. After listening to some of their answers, I will explain how friction from the tracks and air drag slow down the ball:

U_i + K_i = U_f + K_f + Heat

U_i + 0 = 0 + K_f + Heat

mgh_i - Heat= 0.5mv_f^2

mgh_i > 0.5mv_f^2

v_f < √(2gh)

Theoretically, the mass of the ball should not affect the final velocity. But it could produce more friction with the tracks. A heavier ball is rolled down the tracks and the measured velocity will be compared to the expected outcome.

Given that the fourth track has a dip, it should challenge the students’ ability to apply the principle of conservation of energy. Specifically, their ability to understand that all that matters when applying this principle is the initial and final height of the ball and not the path it took. Again, the expected and observed outcome will be compared and discussed.

The fourth track, however, will produce the largest amount of friction because the dip will produce some centripetal acceleration, which will increase the normal force and therefore increase the friction between the ball and the track.

An important goal of the presentation is to encourage students to develop their critical thinking skills by asking questions such as: why does the ball have less average final velocity on the third and fourth track than the first two tracks? How does the size of the gap in a track affect the energy of the rolling ball, what does it have to do with rotational energy? How does the distance between the timer gates affect the time measurements?