Rolling Rally

Hypothesis

The ball with the smallest radius will reach the bottom before the balls with the larger radii. Considering that we just learned about torque and what was mentioned about friction causing balls to roll (instead of slide), I figure that the force of friction is realized as torque on the ball. Therefore, a ball with a smaller radius will experience more torque with a similar frictional force and roll down more quickly.

Procedure

1. Obtain various balls of different radii, masses, and materials.
2. Set up two side-by-side inclines
3. Roll two balls down simultaneously, using a straight object such as a ruler to release them at the same time.
4. Record results.
5. Repeat steps 3–5 for additional trials.

Specimens

Spherus metallica

Commonly known as the metal ball, specimens of the species Spherus metallica are extremely shiny, though their luster dulls with age.

radius: 0.016 m
mass: 0.0662 kg

Spherus plastica cavum

Closely related to Spherus plastica solidis, the hollow ball, as this species is known, is noticeably lighter, redder, and squishier than its proverbial cousin.

radius: 0.037 m

mass: 0.0850 kg

Spherus plastica solidus

This medium-sized species is also known as the solid ball and can be recognized by its particularly yellow hue.

radius 0.035 m
mass: 0.1514 kg

Spherus vitrum

Spherus vitra, also known as the glass ball, is unusually transparent for a species in the Spherus genus. 

radius: 0.016 m

mass: 0.0207 kg

Data

First Set of Trials: Spherus metallica vs. Spherus vitrum

11 trials were completed. The common name of the specimen that reached the bottom first is indicated.

1. metal ball
2. metal ball
3. glass ball
4. metal ball
5. glass ball
6. metal ball
7. metal ball
8. metal ball
9. metal ball
10. glass ball
11. glass ball

Second Set of Trials: Spherus plastica cavum vs. Spherus plastica solidus

9 trials were completed. The common name of the specimen that reached the bottom first is indicated.

1. solid ball
2. hollow ball
3. solid ball
4. solid ball
5. solid ball
6. solid ball
7. solid ball
8. solid ball
9. solid ball

Third Set of Trials: Spherus metallica vs. Spherus plastica cavum

7 trials were completed. The common name of the specimen that reached the bottom first is indicated.

1. hollow ball
2. hollow ball
3. both
4. both
5. hollow ball
6. hollow ball
7. hollow ball

Conclusion

Guess what? My hypothesis was incorrect. Here’s my reasoning.

Based on the first set of trials, it appears that mass does not affect the amount of time it takes for a ball to reach the bottom of the ramp when the radii are the same, as neither ball reliably does so first. This makes the second set of trials particularly interesting because it seems that something other than mass is making one of the balls reach the bottom first. And why the solid ball? The velocity of the ball must have something to do with the distribution of matter in the ball. As the two balls in the third set of trials have too many differences, not much can be concluded from this data in particular.

Okay, so how does all of this fit into what’s actually going on? This data is consistent with the idea that there is a rotational analogue to mass, called moment of inertia.

Think about this: a ball that rolls down the ramp faster than another ball will have a greater angular acceleration, and therefore a greater torque, but also a lesser mass. In linear kinematics, this situation can be represented by the formula F = ma, where F is force, m is mass, and a is acceleration. In angular kinematics, the linear units can be replaced with angular units, giving us τ = Iω, where τ is torque, I is moment of inertia, and ω is angular acceleration.

In linear kinematics, increasing the mass yields less acceleration. Therefore, in angular kinematics, increasing the moment of inertia also yields less (angular) acceleration. So in all of these trials, the balls that reach the bottom of the ramp first have a smaller moment of inertia. But this raises the question, “What is moment of inertia‽”. Well, simply put, it’s rotational mass. Objects with different distributions of matter have different moments of inertia. For a solid sphere, such as most of the specimens, the moment of inertia is mr² / 2.  However, a hollow sphere has a moment of inertia of just mr², which is why the results of the second set of trials favored the solid ball.

So, to conclude, it isn’t quite the mass or radius of a ball that determines how long it takes to reach the bottom of a ramp. It’s more like a combination of the two that’s also affected by the distribution of matter in the ball.

Centripetal Force

What’s the Big Idea‽

Does Newton’s Second Law of Motion (ΣF = ma) stand up in rotational kinematics? Let’s find out! In this lab, we’ll be validating the formula ΣF_C = mv² / r, where ΣF_C is the sum of the forces in the centripetal direction, m is the mass, v is the velocity, and r is the radius.

Procedure

This lab seemed simple enough; the idea was that we would construct a pendulum, then measure its centripetal force and linear velocity at a single instance. This absolutely did not go as planned—it seemed like the force probe and the photogate were never working correctly at the same time. However, after several attempts, we somehow managed to get everything in order.

To begin our final attempt, we gathered a stand with a metal rod and clamps to create the structure that the pendulum would be attached to, then used a force sensor, some string, and a metal ball to construct the pendulum itself. At the base of the pendulum, we attached a photogate to measure the velocity of the ball.

Exactly as so!

We connected the force sensor and the photogate to the Vernier LabQuest, profusely checking them to make sure they worked. We also used a meterstick to measure a radius of 0.5 m. Next, we lifted the ball along a circular path and released it, recording the following data:

Though somewhat unrelated, the data from the force sensor approximates a sine curve as it is uniform circular motion 

Less interestingly, the data from the photogate decreases over time as the pendulum slows down.

Hungry for some closure on this lab? Time to see if Newton’s Second Law works in rotational kinematics in the Takeways!

Takeaways

By taking two corresponding points from out two data sets, we can see if the formula ΣF_C = mv² / r, is true. Let’s do some math to see what happens!

And the calculations reveal...

Aha! There is only a 0.002 N difference between the force measured from the sensor and our calculated force, meaning that indeed, Newton’s Second Law does continue to work in this strange rotational land of mystery. Our data only yields a 2.1% difference, which is fantastic, especially compared to some of our other labs. Today we go home in confidence knowing that the universe isn’t, in fact, broken. What relief.