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# Rolling balls on sloped tracks

### Overview

Gravity is a force that pulls us, and everything around us, down towards the ground. Actually, gravity is a force that attracts any two objects. The heavier the objects are, the more gravity wants to pull them together. Also, the closer together the forces are, the stronger the force is. So when two people are standing next to each other, there is actually a gravitational force trying to pull them together! However, this force is so small for things as light as people that we don’t even feel the force. It’s another story for the gravitational force of the earth. The earth is so massive that it’s gravitational force is pretty strong. That’s what keeps us down on the ground. It’s also the same force that makes the yellow Chaos ball fall to the ground when you drop it. In this activity we’re going to explore some of the properties of the gravitational force.

#### Setup

Additional materials needed: ruler, masking tape, and stopwatch

Construct a simple sloped piece of track using three straight long sections of track. When complete, your track should look similar to the one in the diagram below. Activity

#### Stage 1: Drop the Ball

If you’re like me, you probably drop things all the time. Sometimes we do it on purpose, and sometimes it’s an accident, like when you drop a deck of cards all over the floor!

Take one of the yellow balls and drop it on the floor. Drop it close to the floor and then lift it up high and drop it. Watch how it falls. Does it fall in a straight line? It does if you just drop it.

Take the ball and throw it a little bit sideways. Now it doesn’t fall straight down, but follows kind of a curved path. Also notice how the balls speeds up as it falls. Do you think anyone could predict exactly where the ball is going to land when you drop it or throw it? More importantly, who cares?

Who cares? Lots of people do, including you. Have you ever played catch with a baseball or football? After a while, you just kind of know where the ball is going to land, and you get better at catching it.

Believe it or not, people have actually figured out how to predict where things are going to land when they fall. And sometimes it’s important. For example, when old buildings are blown up, people have to calculate exactly where the building is going to fall to make sure it isn’t going to land on anything important. And in the circus when someone is shot out of the cannon, it’s important to make sure they land in the safety net!

#### Stage 1: Roll the Ball

You see things roll every day. When you ride a bike you roll down hills, or when you play soccer you see the ball roll across the ground. We even have bugs that roll called rolly-pollies!

But what’s the one common thing about things that roll? They all roll faster when going downhill, and they all slow down when you roll them uphill! Let’s try this out on the track.

Place the ball on the track and let it go. Watch as the ball rolls down the track. Compare this to when you simply drop the ball. What is different? What is the same? Specifically, think about the following questions, and compare the two cases.

·Does the ball seem to speed up as it falls/rolls?

·In which case does the ball speed up faster?

When you drop the ball it falls straight down, and when you put it on the track it rolls sideways down the track. Gravity is pulling straight down on the ball in both cases, so how can this be?

Adjust the slope of the track so that it is steeper. Roll the ball again and notice any changes. From experience you know that the ball is going to speed up faster. Why do you think this is?

Here’s a nice pictorial way to imagine what is going on. The diagram below shows a ball sitting on a gently sloped track and a ball on a steep track. In each case there is an arrow indicating the direction of the gravitational force. Notice that it is straight down in both cases. Also, the length of the arrow corresponds to the strength of the gravitational force. It too, is the same in both cases.

We can draw this arrow as the sum of two arrows. The first arrow we draw perpendicular to the track. This means that it is pointing straight against the track. This arrow is labeled Fp for perpendicular. The other arrow is in the direction of the track. It is labeled Ft for force tangential. The total force, representing the force of gravity, is labeled Fg. The reason the ball speeds up faster on the steeper track is that the part of the force in the direction of the track is larger. (Ft is larger.) For the gently sloped track, Ft is small. The magnitude of the gravitational force (Fg) is the same in both cases. This means that only a small force is pulling the ball along the track. This is why it speeds up so much slower.

#### Stage 2: Calculating the Average Speed

By now you’ve seen the ball roll down that sloped track many, many times. One question you might have is, “Just how fast is the ball rolling?” Well, this question can be tough to answer. Calculating how fast it is going at every moment along the track is pretty complicated. And while it is possible to calculate what it should be theoretically, it is impossible to measure the speed at every spot along the track.

What we can do, though, is to measure the average speed of the ball. The speed of the ball is the distance it travels in a given time. You’re probably familiar with speeds in miles per hour. Notice that this is a distance, miles, in a unit time, an hour. We are going to calculate the speed of the ball in centimeters, cm, per second, s.

The formula for the average speed is the total distance divided by the total time:

s = d/t

In order to measure the speed, we need to measure the total distance the ball travels, as well as the total time. First, place two small pieces of masking tape near the beginning and end of the track. These will be our starting and ending lines. (If you don’t have masking tape, use something else to mark your starting and ending lines.) Next measure this distance as accurately as possible. Your ruler has both centimeters and inches. Make the measurement in centimeters and record it in the chart below. For this activity the distance will be the same for each trial, so put this number in each row of the distance column.

Trial Number          Distance          Time              Speed

1

2

3

4

5

Average Speed =

Compute the average speed by adding all five speeds together and dividing the total by five.

Now it’s time to make the measurements. For this you will need a stopwatch. Place the ball at the starting line and hold it there stationary (not moving) with your finger. Start the stopwatch at the same time as you let the ball go. Stop the stopwatch when the ball crosses the finish line and record your time in the chart. Repeat until you have done this five times. Then calculate the average speed.

Note: If you’re having trouble getting accurate times, you might consider making the slope gentler. This will cause the ball to roll slower and it will take longer to reach the end. Having longer times will help you get a more accurate measurement of the speed.

The reason we take five measurements and then use the average as our answer is that if we just make one measurement, we are more likely to make a mistake. If you look at your five speeds and can see that one of them is drastically different than the others, you know that you probably messed up on that one. And in general, sometimes you will measure the speed to be a little too fast, and other times you will measure it too slow. By averaging the results, you will get a number that is closer to the actual average speed.

Scientists and engineers use this technique all the time. Rarely do they perform an experiment only once. Usually they perform several experiments to make sure they are confident of the results.

In this experiment, the average speed is the speed the ball is traveling when it is in the middle of the track. Before this point, the ball is traveling slower, and after this point it is traveling faster. For example, at the beginning of the track the ball isn’t moving at all, and at the very end the ball is moving twice as fast as the average speed!

In the following activities we’ll make some changes and see how this affects the results.

#### Stage 2: Dependence of Average Velocity on Slope and Overall Height

Here we’re going to repeat the above activity for several different slopes. We’ll start with a very gentle slope and gradually work up to a pretty steep one. Set up your track so that the slope is very gentle. The total vertical drop over the length of the track shouldn’t be more than an inch or two. Your track should look like this. h=Total Vertical Drop
d=Total Horizontal Distance

Now we’re going to measure the slope. In order to measure the angle of the slope we need to know the total horizontal distance and the total vertical distance of our track.

Carefully measure the horizontal and vertical dimensions of your setup using a ruler or tape measure. Be as accurate as you can in this measurement. We want to use these two numbers to determine the angle of incline of the track. The tangent function does this for us. If you’re not familiar with i, most calculators have a built in function that can compute the angle of the track given these two numbers. This function is the tangent function and is defined as:

tan = vertical distance / horzontal distance

Most calculators have this function built in and will compute this for you. You can either give it an angle and it will give you the value of the tangent of the angle, or you can give it the value of the tangent and it will give you the corresponding angle. We are going to compute the value of the tangent and ask the calculator to give us the angle. Fill in the values in the bottom two rows in the table and use your calculator to compute the value of the angle. If you don’t know how to use the ArcTan function on your calculator, ask a friend, teacher, or parent.

Trial Number          Distance          Time          Speed

1

2

3

4

5

Average Speed =

Compute the average speed by adding all five speeds together and dividing the total by five.

Vertical distance =

Complete the rest of the table as you did in the previous activity. Basically, measure the average speed of the ball for five trials.

Next, change the angle so that it is a little steeper and fill in the table again. Do this for five different angles.

The last thing we want to do is to compare the results for the different angles. What do you think we’ll see?

To see what happened, plot the vertical distance versus the average speed on the diagram below.

You can see that the average speed increases for increasing height. Well, that’s probably what you expected. The ball is falling from a greater height, so it has more kinetic energy (energy of motion) by the time it gets to the bottom. This means that the ball goes faster when we let it fall from a greater height.

You may have thought that plot of points would be a straight line. It seems like if the ball starts out twice as high, it should be going twice as fast when it reaches the bottom. Well, it is going faster, but not twice as fast. In order for the ball to go twice as fast, we need to drop it from four times the height! That may seem a little weird, but that’s just the way it is. A few centuries ago, Sir Isaac Newton was the first person to understand why this is.

#### Other activities to follow using a similar setup

Repeat the above activity for several different slopes of the track. For each slope, calculate the total height the ball falls and the average speed of the ball (using the method above.) Make a plot of the height vs. the average speed. What do you notice about the points of the plot. The average speed increases with increasing height. (Although it is not a linear relationship.)

Use this activity to introduce the concept of energy and that average speed of the ball a representation of its kinetic energy.

The next activity will be to again use several different angle, but now we will also change the distance the ball goes to ensure that the ball falls the same height each time, regardless of the angle. Again, for each angle we will compute the average speed. Then, we plot average speed vs. angle. This time the height is fixed. What will this plot look like? (The average speed is the same, regardless of the height.) This further illustrates the idea that the speed of the ball is related to the vertical distance it has fallen. This distance is related to the change in potential energy of the ball.

The concepts of kinetic energy, potential energy, and conservation of energy will be explored in further activities. We could make an extension of this activity for more advanced students where they calculate the kinetic energy of the ball using the average speed. They could relate this to the change in potential energy.

### Quiz Yourself

#### Stage 1

You are invited to go skiing. At the top of the hill, you see three signs: Easy, Intermediate and Expert. You tell your guide that you don’t want to go too fast, and he says that the least steep hill is the Easy, and the steepest is the Expert. Being concerned with how fast you’ll go on the hill, which trail do you go on. How is your speed related to the steepness of the hill?

If you want to go slow, you choose the Easy trail.  The more difficult the trail, the steeper it will be.  The steeper the trail, the faster you go.

#### Stage 2

Suppose that in the first exercise the intermediate trail has an angle of 30 degrees.  How strong is the component of the gravitational force pulling the skier forward?  The skier weighs 50 kg.

The component of force pulling the skier forward is:

Ft = Fg sin(theta) = mg sin(theta) = 50kg 9.8m/s² sin(30) = 245kg m/s² = 245 N

### Chaos has enjoyed a long history of continuous recognition and awards. 