Elevating

Elevating, Episode 1: What’s the Big Idea‽

December 22^nd, 2014—inhabitants of Grafton High have recently reported bizarre phenomena occurring at the local elevator. Several accounts describe unusual feelings of “weight shifts”, noting altered senses of weight while in transport. The reports have prompted unease in the community of scientifically-literate students who understand weight as being the gravitational force on any given object. A local group known only as the Super Six have attempted to explain the rumors, suggesting acceleration as a possible cause of these strange symptoms.

As a credible news source, we are obliged to tell you this image was retrieved here.

As the Super Six begins research, they realize that all their findings point to one culprit: their archvillain, AP Physics!

Elevating, Episode 2: Procedure

To investigate the correlation between perceived weight and the elevator’s acceleration, the Super Six return to their super-secret hideout to obtain the proper equipment to conduct their experiment: the Fantastic Force Plate and their truly secret weapon, the Vernier LabQuest. They rush to the elevator and, using their extraordinary Physics Powers, immediately configure the Fantastic Force Plate with the Vernier LabQuest to prepare for the real action.

For a second, silence. The enter the elevator cautiously, under the guise of civilians to avoid the compromise of their true identities. And, as a completely reputable news source, we have last-minute photographs capturing the Super Six in their attempts to fight crime and conquer AP Physics.

Two members of the Super Six—codenamed “Sam” and “Owen” apply their weight to the Fantastic Force Plate to measure the force they exert. Of course, being the amazing superhero team they are, the Super Six selected the members with the greatest deviation of masses to thwart AP Physics and minimize potential error. The elevator accelerates, but then—BAM!—AP Physics sneaks up on the entire Super Six! Weight-shifting ensues, as shown in our exclusive video clip below.

Clearly, the Super Six isn’t kidding around when it comes to justice, but AP Physics isn’t about to give up. Its anguish-inducing weight-shiftings torment them mercilessly! Upon (upwards) acceleration, they feel heavier, but as the elevator reaches its maximum velocity, the weight-shiftings stop, only to afflict them again upon deceleration, when they feel heavier. What will AP Physics do next‽ Will the Super Six survive this torture‽ Find out next time!

Elevating, Episode 3: Takeaways

Using their super secret weapon, the Vernier LabQuest, the Super Six escape the grasp of AP Physics just in the nick of time! They peruse their collected data, probing it for answers to the unexplained weight-shiftings.

Super Six Archive 98183447: Owen’s Data

Super Six Archive 76117103: Sam’s Data

Super Six Archive 57640556: Sam and Owen’s Data

“Aha!” someone exclaims. The Super Six finally notice that an increased force—caused by the upwards acceleration of the elevator—felt like an increase in weight, i.e. the purported “weight-shifting”. Likewise, a decreased force felt like a decrease in weight. Of course, they determine, these events occurred in a different order depending on the initial direction of the elevator. They realize AP Physics’ ploy all along—the sense of weight isn’t just affected by the gravitational force on an object, but the other forces as well! They return their results to the esteemed Dr. J. E., who succinctly details the specifics of the situation:

Courtesy of Dr. J. E.

The Super Six relate their conclusion back to the equation F = ma. The total force in the vertical direction is what is perceived as weight, regardless of the gravitational force. In fact, they realize, that the formula can be rewritten as F_G + F_E = m(a_G + a_E), where the superscript G denotes gravity and the superscript E denotes the elevator. When the elevator’s acceleration is downwards, the force is subtracted from the total force, making it seem like there is less weight, and more weight in the opposite case.

The Super Six finalize their conclusion, but then—SCREAMING! Horrible, ear-piercing cries for help. The Super Six hear a group of people shrieking from injustice, presumably AP Physics out on another weight-shifting spree. They pinpoint the sounds to the nearest amusement park, only imagining what wrongdoing their archvillain could cause on a ride with an intense downwards acceleration. “Weightlessness” is the word that comes to their minds as they sprint into the distance.

⁂

It’s nighttime. The Super Six have relinquished the rest of their energy, back at their super-secret hideout. Chinese food occupies the entire table, and all they do is eat. Oh wait—takeaways? Eh, oh well. Close enough.

Newton’s Third Law

What’s the Big Idea‽

N3L. Newton’s Third Law. Though we’re quite familiar with the idea that “For every action, there is an equal, but opposite reaction”, do we really know what it means? Time to investigate, to determine what Newton’s Third Law looks in a real, physical setting!

Procedure

For this lab, we used a Vernier LabQuest, a dynamics track, two (relatively) frictionless carts, two force sensors, and three different kinds of connectors. First we placed the track flat on the table and used the iPad to measure how level it was, adjusted its height as needed. We placed two dynamics carts on the track, each having a force sensor (one labelled Sensor A and the other Sensor B) on top of it, but facing towards each other. Next, we calibrated the force sensors as necessary and zeroed their values when they were experiencing minimal force. For one of the force sensors, we reversed the sign recorded values to account for the difference in direction.

In total, we performed 12 trials plus an extra trial just for the heck of it. The first four trials investigated two different methods of moving the carts and two different types of connectors. During the next four trials, we repeated these same steps but with an additional mass of 500g on the cart with Sensor A on it. The propreantepenultimate, preantepenultimate, antepenultimate, and penultimate trials primarily dealt with pushing the sensors towards each other in lieu of using a connector (bumpers were placed where the connector would have been). And finally, the very last trial had no major goal and used a spring as the connector between the carts. 

Above shows various setups for different trials. Counterclockwise from the top right: an additional mass on the cart; the carts attached by string; the Vernier LabQuest; the carts attached by a rubber band. Here’s a general overview of each trial:

Trial	Connector	Method	Additional Mass (Sensor A)
1	string	pull on Sensor A and Sensor B	0g
2	string	pull on Sensor A	0g
3	rubber band	pull on Sensor A and Sensor B	0g
4	rubber band	pull on Sensor A	0g
5	string	pull on Sensor A and Sensor B	500g
6	string	pull on Sensor A	500g
7	rubber band	pull on Sensor A and Sensor B	500g
8	rubber band	pull on Sensor A	500g
9	none	push Sensor A and Sensor B towards each other	0g
10	none	push Sensor A towards Sensor B	0g
11	none	push Sensor A and Sensor B towards each other	500g
12	none	push Sensor A towards Sensor B	500g
13	spring	none in particular	0g

The following images show the data for each trial sequentially.

Wow, that’s a lot of data. The thing is, it all boils down to the same conclusion!

Takeaways

The consistency among the thirteen trials, even the last one, is remarkable. To reiterate Newton’s Third Law, “For every action, there is an equal, but opposite reaction”. That shows itself quite clearly here—the data collected by the force sensors is is the same for each, just in the opposite direction! It didn’t matter if we changed the cart that was being pushed or pulled, the type of connector we used, or even the mass of one of the carts. Isn’t physics fantastic?

Atwood’s Machine

What’s the Big Idea‽

Ultimately, the goal of this lab is to answer three questions:

1. What are the variables that determine a system's state of motion (constant velocity vs. constant acceleration)?
2. What happens when the total mass of a system is kept constant, but the net force is varied?
3. What happens when the net force on a system is kept constant, but the total mass is varied?

Along the way, we’ll be discussing Newton’s laws of motion, inertia, equilibrium and disequilibrium, mass, velocity, acceleration, and force.  Luckily, we have the perfect tool to answer these questions: an Atwood’s Machine!

Procedure

However, before we can actually answer these questions, we need to make an Atwood’s machine. Turns out it wasn’t too difficult, at least this time. I can safely say that no major injuries were sustained*.

Fig. 1 Our beautiful Atwood’s machine.

Basically, we attached a photogate (at the top) to a metal rod, which supported the entire structure. In between the photogate’s sensors was a pulley with masses on both ends. To investigate equilibrium and disequilibrium, we could add and remove masses from each side.

Fig 2. Closeup of the pulley.

Fig. 3 Closeup of the masses.

But what would a lab be without Logger Pro? We used Logger Pro to take the data from the photogate and convert it into the velocity and acceleration of the pulley.

Fig. 4 QR code on Logger Pro used to log data on our iPads.

With everything set up, now to answer the first question!

1. What are the variables that determine a system's state of motion (constant velocity vs. constant acceleration)?

To answer this question, we experimented with directly applying force to one side (by tapping it) versus constantly applying a force (by adding a mass). For a nice visual, the latter looks something like this:

Fig. 5 Placing a 2g mass.

We completed three trials using this method. The first trial consisted of pushing one side of the Atwood’s machine, whereas the other two trials consisted of placing additional weight on it.

	total m	net m	side 1 m	side 1 Δm	side 2 m	side 2 Δm
Trial 1	100g	0g	50g	n/a	50g	n/a
Trial 2	102g	2g	52g	2g	50g	0g
Trial 3	105g	5g	55g	3g	50g	0g

Below shows Logger Pro’s output of the three trials, analyzed in the takeaways.

Fig. 6 Trial 1

Fig. 7 Trial 2

Fig. 8 Trial 3

On to the second question!

2. What happens when the total mass of a system is kept constant, but the net force is varied?

This question can be answered using the same strategy as the first, but with alterations. The video below shows what I’m talking about.

Fig. 9 Explosion of masses optional.

We performed three trials similar to the above, augmenting the mass by ±10g each trial.

	total m	net m	side 1 m	side 1 Δm	side 2 m	side 2 Δm
Trial 1	150g	50g	100g	n/a	50g	n/a
Trial 2	150g	30g	90g	10g	60g	10g
Trial 3	150g	10g	80g	10g	70g	10g

Each trial yielded telling results, once again, analyzed in the takeaways.

Fig. 10 Trial 1

Fig. 11 Trial 2

Fig. 12 Trial 3

Hark! Question 3 awaits!

3. What happens when the net force on a system is kept constant, but the total mass is varied?

This question is very similar to the last, so it only requires minor variations in the masses placed on each side. The difference in the masses of each side remained the same, but we consistently increased the amount of mass each trial.

	total m	net m	side 1 m	side 1 Δm	side 2 m	side 2 Δm
Trial 1	150g	50g	100g	n/a	50g	n/a
Trial 2	190g	50g	120g	20g	70g	20g
Trial 3	230g	50g	140g	20g	90g	20g

Fig. 13 Trial 1

Fig. 14 Trial 2

Fig. 15 Trial 3

Alright, so that’s the last of the data, time to delve into it!

Takeaways

As a refresher, below are the questions we attempted to answer with our three experiments of three trials each.

1. What are the variables that determine a system's state of motion (constant velocity vs. constant acceleration)?
2. What happens when the total mass of a system is kept constant, but the net force is varied?
3. What happens when the net force on a system is kept constant, but the total mass is varied?

Let’s look at the first one. This question directly relates to Newton’s first law of motion, which states:

“An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force” (Retrieved from the Physics Classroom).

Turns out, our first question was discussing inertia: the tendency of an object to remain in the same state of motion. We noticed that, when the two sides were in equilibrium (i.e. they had the same amount of mass), they remained at rest. However, upon applying a one-time force (such as a push), the two sides moved at a constant velocity (see Fig. 6). Since the forces were still balanced (after the initial push), we noticed that it continued at the same, consistent velocity. But upon applying a constant force that would unbalance the entire system, we observed different results (see Fig. 7 and Fig. 8). A difference in mass of the two sides causes one side to accelerate downwards and the other side to accelerate upwards. Weight is effectively the force of an object due to gravitation, and since the masses were unequal, the entire system was in disequilibrium. And, when the weight difference was greater, so was the acceleration.

As for the second and third questions, we’re looking at Newton’s second law of motion, which essentially states that force is equal to the mass of an object times its acceleration. In designing an experiment to answer the last two questions, we were able to explore this. When the total mass of a system is kept constant, but the net force was varied, we noticed that different trials (see Fig. 10, Fig. 11, and Fig. 12) yielded different accelerations, suggesting that there is a direct correlation between mass and force as well as between acceleration and force. When the net mass or the acceleration increases, so does the net force  (see Fig. 13, Fig. 14, and Fig. 15). When the net force is kept constant, but the total mass is varied, the trials still show a change in acceleration, due to the increased mass of the entire system.

*I just put that asterisk there to make it sound like injuries actually were sustained. But no, there weren’t any (at least that I was aware of).

More Projectile Motion

What’s the Big Idea‽

So now that we understand the basics of projectile motion, how can we use that knowledge to calculate how a projectile will behave? We can design a setup that allows us to determine the preexisting conditions of a launched projectile, which we can then use to calculate where the projectile will land. After that, further inquiry will allow us to delve into advanced topics dealing with different angles and the location of the projectile-launching mechanism. 

Procedure

Essentially, all we needed to do was design something that would allows us to launch a small metal ball horizontally, with a photogate on the end to measure the velocity of the ball. Instead of just propping the ramp, we used a clamp to attach it to a metal rod. The drama.

The above photograph shows our setup, with the ramp suspended in the air and a photogate at the end (which is attached to a Vernier LabQuest). It was a bit unstable, so we later added a book underneath the bottom of the ramp to keep its height from varying. However, now we needed to worry about how we were going to go about measuring the velocity of the ball as it leaves the ramp. Using a meter stick, we measured the width of the ball (2.3 cm); the photogate allows us to record a single velocity, taking into account how far the ball moves when it interrupts the light (i.e. the ball’s width). However, we noticed that this method of recording the ball’s velocity is prone to error—what happens if the ball blocks the photogate at a height that is to high or to low? Since it’s spherical, the time it takes will appear to be reduced, giving us a faulty velocity. The solution? Lasers!

Above is super awesome laser.

To ensure that the photogate is aligned with the ball, we used a laser to measure the distance of the midpoint of the ball from the top of the table, and adjusted the height of the photogate likewise.

After configuring the LabQuest, all we needed to do was let the ball roll down the ramp, as shown in the video above. To reduce random error, we repeated this process multiple times, recording the velocity each time. To reduce systemic error, we made sure to consistently place the ball at the very top of the ramp.

A screenshot of the LabQuest shows us one of ten trials, our velocities as follows:

• 1.616 m/s
• 1.586 m/s
• 1.603 m/s
• 1.593 m/s
• 1.585 m/s
• 1.619 m/s
• 1.579 m/s
• 1.615 m/s
• 1.624 m/s
• 1.618 m/s

Averaging this data yields a value of 1.604 m/s, our horizontal velocity. We also measured the vertical distance (the height from the ground to the “launch point” of the ball), which was approximately 115 cm. Let’s use our physics knowledge and some kinematics equations to find the horizontal distance the ball travels!

1. First, we’ll need to know how much time it took the ball to fall. We can determine this using the equation Δy = v_iyt+ ½gt².
2. We know what the values of Δy (vertical displacement), v_iy (initial velocity in the y-direction) and g (the force of gravity) are. Substituting them in, we get -1.15 = 0 × t + ½ × -9.8 × t² which simplifies to -1.15 = -4.9t².  Rearranging the terms and simplifying them yields an approximate value of 0.484 s for t.
3. Knowing t, we can now solve for the horizontal distance using the formula v_x = Δx ÷ t. Quickly inputting the values for v_x and t yields a value of about 0.776 m for Δx.

Hooray! This value is very close to the actual horizontal distances we recorded—67 cm, 69 cm, and 71 cm in three independent trials using an identical setup as above. Turns out we can use kinematics to our advantage!

Unfortunately, being in a two-person lab group and running out of time to develop new physics contraptions means that we weren’t able to explore much more in this lab. However, we were able to at least start the next part of the experiment. Above, we fashioned an adjustable ramp using various rods and clamps, allowing to alter the angle of the ball, and thus, its trajectory. We recorded a few trials using a single angle, measuring the ball’s velocity a few times. However, this data has escaped me, and it wouldn’t be of much use anyways seeing as we were unable to finish the portion of the lab.

Takeaways

Though I explained a lot of this above, the first portion of the above allowed us to use the skills we’ve learned previously and apply them to calculate a specific value (namely the horizontal displacement) and then check to see if that value was indeed correct. Which it was! It’s fascinating and beautifully concrete when we can perform and experiment and have the data prove itself.

There were a few questions that this lab left unanswered, unfortunately, mainly because of time restrictions. Though I don’t have the data to back it up, I’ll answer the questions regardless.

• When launching a projectile at an angle, what angle (or angles) produce the greatest range? A 45° angle will produce the greatest range when launching a projectile. This makes sense intuitively, because an angle that is too high will cause the projectile to move a lot in the vertical direction, but ultimately, very little in the horizontal direction. And of course, gravity always gets you down. A range that is too small will not allow the projectile to move far enough because it will hit the ground to early.
• How do the equations developed in the earlier part of the lab need to be modified to deal with projectiles that are not launched horizontally? The equations themselves don’t actually need to be modified—all you need is a bit of trigonometry! By breaking up the initial velocity into horizontal and vertical components, physics continues to work as expected.
• Move your set-up onto the floor. Does this move impact the angle that gives you the maximum range? Though we didn’t do this, it shouldn’t impact the angle that gives us the maximum range. All it does is reduce the amount of time before the ball hits the ground, since the vertical displacement is shortened.
• How does the range of projectiles (across level surfaces) differ for steeper or shallower launch angles? I kind of explained this in the first question, but I can elaborate a bit. As the angle nears 45°, the range of the projectile increases, and likewise, as the angle nears 0° or 90°, the range of the projectile decreases.

The end! For now, at least, for now.

Projectile Motion Video Analysis

What’s the Big Idea‽

Projectile motion occurs when an object simultaneously exhibits both horizontal and vertical motion. Although these motions occur at the same time, they are considered independent. But how can we prove this? In this lab, we use Logger Pro to graph a video of a ball in projectile motion to see how it behaves in the x- and y-directions.

Procedure

The first part of the lab consists of getting our equipment ready and recording an example of projectile motion. To perform the first part of the experiment, we need a meter stick, an iPad, and a ball. First, we lined up the meter stick to the wall to set the scale of the video and prepare it for the video analysis, as shown below.

With that completed, it’s time to film! Projectile motion involves both vertical and horizontal movement, so we throw the ball into the frame of the camera at an angle. Here’s the result:

The second part of the lab is where it gets interesting. After loading the video into Logger Pro, we set the scale of the video, using the measurements provided by the meter stick as a guide. Then, going frame by frame through the video, we plot the center of the ball, as shown below:

Afterwards, we can view the position vs. time graphs. The red dots represent the change in horizontal position and the blue dots represent the change in vertical position.

The shapes of the graphs are the most telling: the change in horizontal position is clearly linear, and the change in vertical position is clearly quadratic. This tells us that no new forces are acting upon the ball in the x-direction and that one consistent force is acting upon the ball in the y-direction, resulting in a gradual change in its velocity.

If we had the velocity vs. time graph, we’d notice that the x-direction would show a completely horizontal line and the y-direction would show a linear graph, indicating a constant acceleration.

Takeaways

So what does all this data mean? What does it show? We know that the original force and velocity can be split up into two separate pieces, one in the horizontal direction and one in the vertical direction. The velocity in the horizontal direction remains constant for the duration of the ball’s flight, however, the velocity in the vertical direction changes, in fact, at a constant rate of -9.8 m/s. Gravity causes the acceleration of the ball downwards, but it only affects it in the y-direction, not in the x-direction as well. These two motions are effectively separate.

However, there are some things that we could have done better in our experiment. Firstly, it would be helpful to record the video a little farther back to see where the ball starts, just for some context. Additionally, having the meter stick in the same plane as the ball while the video is recording (as opposed to measuring the wall) would help eliminate potential sources of error in our graphs. Ultimately, this resulted in strange values for distance in our graphs, none of which made sense. Lastly, we should have saved a version of the graph showing velocity vs. time as well because it would aid in our understanding of projectile motion.

Determining g: Evaluating Methods of Measuring Acceleration

Please ignore the bouncing title. Like seriously. It’s not even proper HTML.

What’s the Big Idea‽

How does gravity affect the motion of objects? The force of gravity is consistently affecting everything on our planet, but its effect becomes even more apparent in falling objects. Gravity causes all objects to undergo a constant acceleration of 9.8 m/s²—but how can we calculate this value (notated as g) using only our knowledge of kinematics? 

We completed two different labs to find out and then determined which method was better.

Procedure

The first lab was the Picket Fence Free Fall lab, which required a photogate, a LabQuest, and, of course, a picket fence (which is a thick, transparent strip of plastic with black rectangles evenly space apart). There were also some additionally materials such as the stand and the cloth to make for a soft landing. The resulting setup looked like this:

Once everything was connected, we dropped the picket fence as indicated by the diagram. One of our trials looked like this:

Notice the quadratic curve in the distance-vs.-time graph, a hallmark of constant acceleration. The velocity-vs.-time graph is linear, because the velocity increased at a constant rate. We repeated the procedure several times to get the following composite graph:

Each trial resulted in similar graphs that matched the first.

Now, on to the second lab: the Ball Toss lab! The materials were simple: a ball, a motion detector, and once again, a Vernier LabQuest.

The procedure, like the materials, was also simple: just hold the ball in front of the motion detector, throw it up, and catch it before it hits the motion detector. However, I found this to be rather clumsy because it was difficult to throw the ball straight up and have it return to the starting position. Our most successful trial yielded the following graph:

Like the first lab, we can see the quadratic curve in the first graph and the linear one in the second—giving us insight into how gravity affects an object. The velocity graphs of both experiments happen to be particular insightful—notice the value for m. The first experiment yields a value of 9.666 and the second experiment yields a value of -8.953. Though the signs of these values aren’t important, their magnitudes are. Roughly speaking, they match the Earth’s value of acceleration due to gravity—9.8 m/s² downwards.

Takeaways

If you haven’t realized it yet, we performed these experiments on Earth. Meaning our predicted values for the acceleration due to gravity were fairly accurate, however more so in the Picket Fence Free Fall lab. These predicted values are the criteria by which we are judging the two labs.

There are a couple reasons why I think the first experiment resulted a value closer to the one we were looking for. This concerns accuracy and precision. Accuracy is generally associated with systemic error, whereas precision is generally associated with random error. We were able to eliminate much more (but certainly not all) systemic and random error in the Picket Fence Free Fall lab, meaning our value of 9.666 was was both more accurate and more precise than the one we got in the Ball Toss lab. We were able to repeat the first experiment multiple times without changing the result very much, which ultimately allowed us to average our data (minimizing random error and increasing precision). Furthermore, dropping the picket fence was much less awkward and clumsy than throwing the ball up, which explains the difference in accuracy (the first experiment was affected by less systemic error than the second lab).

Ultimately, I felt that the tools (i.e. the photogate and the motion detector) were both fairly reliable and not a major source of error in either experiment.

And for gravity? Well, we were able to visualize its affects on an object in free fall graphically, as well as roughly determine its value based on our knowledge of kinematics.