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.

Impulse and Momentum

What’s the Big Idea‽

This lab explores the Impulse-Momentum Theorem and the relationship between impulse and momentum.

Procedure

On the technical side of things, we needed a Vernier LabQuest, a laptop, a motion detector, and a force probe. On the less technical side of things, we needed a dynamics cart, a frictionless track, a 500 g mass, some string, a rubber band, and a rod with its accompanying stand.

To set up, we placed the cart on the track, with the motion detector on one side and the force probe secured on the rod on the other side. The end of the cart was attached to a rubber band which was attached to the string which was attached to the force probe.

The basic setup. 

Another view of the setup.

After setting everything up, we measured the mass of the cart (which was 0.51 kg) and opened Logger Pro. We calibrated the force probe and began collecting data, pulling the cart away from the force probe and letting it bounce back. The combined use of the force probe and the motion detector allowed us to measure position, time, velocity, and force, as shown in this graph:

The measured data.

What’s fantastic about Logger Pro is that it allows us to easily calculate and visualize the impulse given to the cart by taking the integral of the curve.

Takeaways

From this lab, we learned that impulse is not only equal to the change in momentum, but the integral of force over time. Additionally, specifically regarding this experiment, a higher spring constant in the rubber band yields a greater change in momentum and therefore a greater impulse.

Momentum and Collisions

What’s the Big Idea‽

The following lab focuses on…you guessed it, momentum! And to that end, we’ll also be playing with the momentum of a system before and after internal collisions. Stay tuned and we’ll perform experiments that not only answer the questions “What is momentum?” and “How is it calculated?”, but also demonstrate conservation of momentum.

Procedure

Like many of our labs, this lab revolved around the low-friction dynamics cats.

That’s dynamic carts, not cats.

We set up two dynamics carts on each side of the track, behind each of which was a motion detector. As usually, a Vernier LabQuest recorded the data onto a laptop. The following photos demonstrate this described setup:

Say hi to Jill!

The motion detectors recorded how the carts’ velocity changed over time.

In addition to using the motion detectors to record the carts’ velocity, we used a scale to measure the carts’ mass.

Cart 1 Pushed; Cart 2 Not Pushed
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.13 m/s	0.07 N/s	0.51 kg	-0.00 m/s	-0.00 N/s	1.03 kg	–	0.07 N/s
during		0.12 m/s	0.06 N/s		-0.00 m/s	-0.00 N/s		–	0.06 N/s
after		-0.00 m/s	-0.00 N/s		0.10 m/s	0.05 N/s		–	0.05 N/s

If we divide the difference of the final momentum and the initial momentum by the average momentum, we can find the percent difference. The first trial yields a percent difference of 15.4%, which while not fantastic, does suggest the conservation on momentum. The higher-than-expected percent difference is most likely due to precision errors incurred when rounding the measured values.

Cart 1 and Cart 2 Pushed at Same Velocity
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.12 m/s	0.06 N/s	0.51 kg	-0.13 m/s	-0.07 N/s	1.03 kg	–	-0.01 N/s
during		0.05 m/s	0.03 N/s		-0.01 m/s	-0.01 N/s		–	0.02 N/s
after		-0.09 m/s	-0.05 N/s		0.12 m/s	0.06 N/s		–	0.01 N/s

Similarly to the last trial, we can find the percent difference. However, this trial yields bizarre results because the average momentum is zero, making it impossible to find the percent difference.

Cart 1 and Cart 2 Pushed at Different Velocity
	Cart 1			Cart 2			Total
	mass	velocity	momentum	mass	velocity	momentum	mass	velocity	momentum
before	0.52 kg	0.40m/s	0.21 N/s	0.51 kg	-0.30 m/s	-0.15 N/s	1.03 kg	–	0.06 N/s
during		0.39 m/s	0.20 N/s		-0.29 m/s	-0.15 N/s		–	0.05 N/s
after		-0.24 m/s	-0.12 N/s		0.33 m/s	0.17 N/s		–	0.05 N/s

The percent difference of the third trial is 18.2%, which is worse than expected. Again, I believe this is due to precision errors.

Takeaways

Although the data isn’t fantastic, it roughly shows us how momentum is conserved with at least some degree of certainty. Before the collision, the total momentum of the system can be calculated as the sum of the carts’ individual momenta (as momentum is a vector quantity). And in the same manner, the total momentum of the system can be calculated to determine the difference between the momenta before and after the collision.

Spring Constant Lab Challenge: SELF DESTRUCT INITIATED

Secure Briefing

Is it just me or does this make it look like physics is espionage?



Especially that last line. The drama.

What’s the Big Idea‽

Excuse me, what’s the big idea‽ How dare you ask such an insulting question on such an important mission. The big idea is that we have a limited time to determine what the spring constant of the dynamic cart’s plunger! There’s no time to waste!

Procedure

First question: what tools do we need to determine the spring constant we’re looking for? Well, we can use the formula F = -kx to do so. Therefore, we need to apply a force onto the cart and measure that force, measure the total displacement that it goes through.

To do so, we used the Vernier LabQuest, a force sensor (with a rubber attachment), a dynamics cart, a meterstick, and a calculator, set up as below:

The calculator’s role was especially important, albeit unorthodox.

Next, we unlocked the plunger and pushed it against the force sensor to measure the amount of force applied, allowing it to go just far enough that the plunger doesn’t lock. A displacement of -0.028 m was measured using the meterstick, with the other data as follows:

The maximum force, as indicated by this graph, is 18.14 N, which corresponds to the maximum compression we obtained above. Next is simple substitution; just use the equation F = -kx and solve for k, then bam! You’ve got a value of 648 N/m for the cart’s spring constant.

Takeaways

A mission well done. We now know that our physics prowess is competent enough to calculate the spring constant of the dynamic cart’s plunger!

Energy of a Tossed Ball

What’s the Big Idea‽

When is energy conserved? When is energy not conserved? And how can we measure that energy? This lab explores these questions by investigating what happens to the energy of a tossed ball.

Procedure

First, we gathered the materials for the lab, which included a Vernier LabQuest and laptop, as usual, but also included a motion detector and a ball (for tossing purposes).

The completed setup.

Once everything was connected, we placed the motion detector flat on the table, tossed the ball directly upwards, and recorded the corresponding data.

The raw data from the LabQuest.

A much clearer readout of the above data.

Oh, and we also measured the ball’s mass with a scale:

mass

0.403 kg

The following table is a synthesis of three recorded points (one corresponding to after the release, one at the top of the path, and one before the catch), yielding data for the time, height, and velocity. This then allowed us to calculate the potential energy (using the formula PE = mgh), kinetic energy (KE  = ½mv²), and mechanical energy (ME = PE + KE).

position	time	height	velocity	PE	KE	ME
after release	1.299 s	0.375 m	2.580 m/s	1.482 J	1.341 J	2.823 J
top of path	1.598 s	0.805 m	0.002 m/s	3.182 J	0.000 J	3.182 J
before catch	1.865 s	0.464 m	-2.580 m/s	1.834 J	1.341 J	3.175 J

If the tossed ball demonstrates conservation of energy, only conservative forces (e.g. gravity) are acting on the system. Otherwise, a combination of conservative (again, e.g. gravity) and nonconservative forces (e.g. friction) are acting on the system. The calculated change in mechanical energy suggests conservation of energy, however, I think that the recorded point directly after release is incorrect. The change from 2.823 J to 3.182 J would only occur if an upward force was temporarily acting on the ball (such as a hand), therefore, this point is unsupportive of our conclusions. (However, in a way, it does show that the hand pushing up on the ball is somewhat like a nonconservative force, because it only acts in one direction.)

Ignoring the first data point, the change in mechanical energy from 3.182 J to 3.175 J makes much more sense. For the most part, the total energy of the system remains constant, suggesting that the most significant forces acting on the system are conservative. I presume the slight decline in energy can be attributed to friction in the form of air resistance (a nonconservative force) acting on the ball.

Takeaways

In this lab, we learned about conservative and nonconservative forces. Basically, as I just described, we can determine whether or not a system demonstrates conservation of energy by measuring its change in energy over a period of time during which forces are acting on it. If the change in energy is minimal, work that displaces an object is done by conservative forces. Contrastingly, if the change in energy is significant, this displacement is partially being done by nonconservative forces.

Work & Energy

What’s the Big Idea‽

We know what work and energy per their colloquial definitions, but what do they mean in physics lingo? This lab explores the relationship between work and different forms of energy.

Procedure

This lab was like three mini-labs wrapped up into one big lab. So that means three mini-procedures!

Mini-Procedure #1

For the first of three mini-procedures, we will need:

• a 1.0-kg mass;
• a meterstick;
• a force probe;
• a Vernier LabQuest;
• a laptop.

Essentially, our objective was to do work on the mass and measure its energy. To do so, we attached the force probe to the top of the mass and lifted it approximately a meter, as performed in the following video:

We averaged the force recorded by the Vernier LabQuest and measured the mass’ displacement with the vertically placed meterstick.

Lifting a Mass
Trial	Mass	Average Force	Displacement	Work
1	1.0 kg	9.75 N	1.00 m	9.8 J
2		9.69 N	0.79 m	7.7 J
3		9.86 N	0.80 m	7.9 J

To calculate the amount of work for each trial, we used the formula W = Fcos(θ)d, where W is the work done on the mass, F is the average measured force, θ is angular difference of the displacement and the applied force, and d is the displacement.

We know that the mass goes from having no potential energy to having potential energy (PE) equal to mgh, where m is the mass of the mass, g is the acceleration to do gravity on Earth, and h is the height of the mass. This can also be described using the formula W = PE. Therefore, the work done should be equivalent to the potential energy the mass has after its displacement. In fact, it does! Notice how the average force is roughly equivalent to the acceleration due to gravity on Earth, which is 9.8 m/s². Calculating mgh results in the very same value.

Mini-Procedure #2

For the second of three mini-procedures, we will need:

• a spring;
• a meterstick;
• a force probe;
• a Vernier LabQuest;
• a laptop.

Similarly to the first mini-procedure, we needed to measure force and displacement. However, for this mini-procedure we stretched a spring instead of lifting a weight.

Fig. 1 The setup used to stretch the spring.

Once we had everything set up, we held the end of the spring and stretched it, using the Vernier LabQuest to measure the force and a meterstick to measure the spring’s change in length.

Stretching a Spring
Trial	Maximum Force	Displacement	Spring Constant	Work
1	4.58 N	0.010 m	458	0.023 J
2	7.25 N	0.015 m	483	0.054 J
3	9.96 N	0.020 m	498	0.100 J

Calculation of work was identical to the first mini-procedure, using the formula W = Fcos(θ)d. Also similarly, the spring goes from having no potential energy to having potential energy (PE), but this time it should be equal to 1/2 kx²., where k is the spring constant and x is the displacement. This situation can be expressed as W = PE or Fcos(θ)d = 1/2 kx², allowing us to calculate both the work done as well as the spring constant. Repeating our calculations with all three trials yields consistent results for the spring constant.

Mini-Procedure #3

Unfortunately, the third and final mini-procedure was never completed. However, if we has completed it, we would have needed:

• a dynamics cart;
• a track;
• a force probe;
• a Vernier LabQuest;
• a laptop.

And look too, there’s a lonely empty chart. We would have been able to accelerate the cart towards a force sensor and measure the amount of force it applied, but alas, we ran out of time.

Accelerating a Cart
Trial	Average Force	Displacement	Time	Acceleration	Work
1
2
3

However, we can still explain what we would have been able to do with the data. Again, we would be able to take the average force (one-half of the measured force) and multiply it by the cart’s displacement to get the work done, i.e. using W = Fcos(θ)d to do our bidding. But I think we’re kinder than that. Anyways, we know that the cart starts out with no energy but gains kinetic energy through the application of work, yielding the formula W = KE, which can be reincarnated as Fcos(θ)d = 1/2 mv². This would have allowed us to calculate the cart’s average acceleration and verify the relationship between the work done and the energy gained.

Takeaways

Even without the last mini-procedure, this lab allows us to answer the following Essential Questions:

1. How can you measure the work which is done?
2. How does the work in each case relate to the change in the energy of each system?
3. How can you measure the energy of each system so that it can be compared to the work that was done?

So, what did we learn?

1. The work which is done is equivalent to a force applied at a distance. Therefore, it can be measured by taking the product of the average force and the displacement caused by that force.
2. The change in work directly relates to a change of energy in each system. In the first two mini-procedures, we noticed that work caused an object with no energy to gain potential energy.
3. Since we can use the measurements of the force and displacement of an object, we must use a different method to measure the energy of each system. However, this varies for each system. Gravitational potential energy can be measured using the mass and height above the ground of an object, whereas elastic potential energy can be measured using the spring constant and change in length of the object being stretched. As for kinetic energy, it can be measured using the mass and velocity of the object. In a given system, the work should be equivalent to the change in energy, assuming the its conservation

Well, that’s all for now! 

Hooke’s Law

WTBI‽

Hooke’s Law: an elastic object’s displacement is proportional to its restoring forces. This speed lab examines this proportionality with minimal verbiage.* Two-day labs deserve to be succinct.

Proc.

Equipment: meter stick; two springs; rubber band; and a LabQuest, connected to a force probe, zeroed when horizontal. First, spring_A lines the edge of the meter stick. Its initial length length_i is measured. It is displaced ∆x in five trials, resulting in final length length_f. The force probe measures F. The same trials are repeated for spring_B and elastic. I hesitate to include multimedia; they say, “a picture is worth a thousand words”. But I must succumb, for science:

Mandatory GIF.

Spring stretching.

Force readout.

All data is recorded below.

Trial	Item	lengthi	lengthf	∆x	F	k	k̅
1	springA	6.5 cm	10 cm	0.035 m	-0.04 N	1.14	1.15
2			15 cm	0.085 m	-0.10 N	1.18
3			20 cm	0.135 m	-0.16 N	1.18
4			25 cm	0.185 m	-0.20 N	1.08
5			30 cm	0.235 m	-0.27 N	1.15

Trial	Item	lengthi	lengthf	∆x	F	k	k̅
1	springB	28.5 cm	33.5 cm	0.05 m	-1.9 N	38.0	42.28
2			38.5 cm	0.10 m	-4.0 N	40.0
3			43.5 cm	0.15 m	-6.4 N	42.7
4			48.5 cm	0.20 m	-9.1 N	45.5
5			53.5 cm	0.25 m	-11.3 N	45.2

Trial	Item	lengthi	lengthf	∆x	F	k	k̅**
1	elastic	17.0 cm	22 cm	0.05 m	-2.0 N	40.0	23.8
2			27 cm	0.10 m	-2.9 N	29.0
3			32 cm	0.15 m	-3.7 N	24.7
4			37 cm	0.20 m	-4.3 N	21.5
5			42 cm	-0.25 m	-5.0 N	20.0

The spring constant: k, consistent in an object. To calculate: F = -kx. The calculation is complete, yielding values for k. k is averaged to produce k̅.

TA’s

This speed lab made Hooke’s Law concrete, understandable. The proportional relationship between force, displacement is clear, embodied by the force constant. Generally, the calculated values for k are consistent except for elastic’s trial 1, an outlier caused by error. This consistency exemplifies the validity of Hooke’s Law and introduces the expected invariableness of k.

*I’m serious about this. Removing words is surprisingly time-consuming.
**Trial 1 is omitted.