The Principle of Superposition
The Principle of Superposition
"Did you stack the bags on the cart like I asked you to?"
Applejack trotted outside, the sun glaring at her even from behind her hat. She yawned. She didn't wake up properly until she had been outside to work, and so despite her having been awake for a good hour now, she still felt lazy and slow. She glanced over at her brother. The big pony in question was in the middle of getting some construction supplies ready, in preparation for some of his jobs this afternoon. Big Macintosh was traditionally in charge of loading and moving the heavy objects. He liked to say that doing a long, repetitive and physical task gave him time to think. Just as well: Applejack enjoyed some of the more social aspects of apple farming, and not having to carry logs of wood around gave her time to do just that.
"Eeyup!" Macintosh replied happily.
Applejack grinned. She was all good to go, then.
Today was a good day to sell apples, she thought. Although Applejack found herself selling apples every day, she had a feeling today was going to be particularly nice. It was something about the weather, and the atmosphere. Somehow, the glow of the sun brushing against the ponies' coats with a slight, ever so gentle breeze made ponies want to eat a delicious, refreshing apple, and preferably one purveyed by the Apple Family.
Of course, first Applejack needed to get her apples from the farm into town. She lived a good ten minutes away from Ponyville, so they couldn't easily manage selling their produce on the farm proper. They needed a vending stand, and the only way to get a large quantity of apples from the farm to town was using a cart, and to follow the road from the farm over to Ponyville.
The cart itself was a battered old thing. One of the wheels was very rusty, and it needed to be oiled constantly. Even then, it often ground against the dirt and made rounding turns difficult. It really was quite a bothersome thing, but until they could replace it, what other choice did they have?
Applejack slipped her head into the harness and quickly adjusted the straps with her teeth. She gave the whole thing a pull to make sure that it was stable. Just like every other day, everything worked. It wouldn't fall over on it's own or anything like that. It was time to start her trip.
The patch of ground just outside her farm was very muddy from all the work they did there, their hooves pummelling the grass and plants away, plants that would otherwise keep the earth firm and solid. A little further on, however, and one would find a very effective dirt road that could sustain the weight of the cart quite easily. Getting it there was troublesome, though. It was as if the mud sucked the wheels right down, sometimes.
"See ya'!" Big Macintosh called out as she left. He would undoubtedly be out fixing some of the fences over by the eastern border of the orchard. There was always something, somewhere that broke, Applejack thought. Nothing ever stayed intact for very long out here.
She reached the dirt road, and the wheels began turning quite effectively. The whole thing was still very heavy: apples, when in large quantities, became very heavy, and the cart itself was far from lightweight. Applejack was a tough pony, though. She could certainly handle it until she got to town.
She let her thoughts wander as she made her way along the winding road. There was almost no doubt that she would meet some of her friends in town: at least one of them could always be found going about their business outside, whether it was Twilight out to buy some files and folders, Pinkie looking for ponies to invite to her latest shindig, or Fluttershy escorting some animals through town. There was always somepony to talk to, and Applejack liked that.
Her ears twitched, detecting a strange and suspicious sound.
She was sure she heard something rustle in the trees to her left. She turned her head to try to locate the source of the noise, her eyes detaching themselves from the road. When she turned her eyes back, she realized with a jump that her cart was about to run over a particularly large stone jutting from the road.
The wheel ran over the rock with a dull thud. Suddenly, the effort it took to pull the cart became much greater as the wheel rose up over the stone. Her muscles tensed, and she grunted, gritting her teeth.
Perhaps Big Macintosh had put too much on the cart again, she thought. He was a little too enthusiastic sometimes.
Thankfully, the rock must have dislodged something in the wheel. Suddenly it didn't feel all that rusty any more, and the cart moved smoothly over the ground like it was brand new. Applejack smiled, and egged on by this run of good luck, went a bit faster. The cart flowed easily behind her, as if weightless.
She heard the sound of flapping wings soon after that. She looked up at the skies, expecting to find one of the weatherponies messing with her perfect day. Instead, it was Twilight: apparently these days there wasn't much for the princess to do except keep an eye out for any of her friends. It must be a very nice life indeed, Applejack thought.
"Hi, Applejack," she said, glancing over at the cart behind her. "It looks like you've got... uh, a very heavy cargo, there."
Applejack nodded. "Yup. Them's some nice, big, fat apples. Cart's a bit old, but it get's 'em to town just fine. Although it's been workin' especially well today. Feels nice an' light."
"Ahm... nothing strange at all about them?"
"Nope. Regular apples."
Twilight frowned, a little confused by something, Applejack didn't know what. She landed next to the farmpony and trotted alongside her, becoming more and more amused as time went on. Applejack wasn't sure what Twilight found so strange: after all, she had seen her tow her apple cart before, hadn't she? It wasn't exactly rocket science. Just two wheels and a few sticks.
"I see. Uhm... tell me, how much do you know about... mechanics?" Twilight asked hesitantly, the thin makings of a smile beginning to play on her lips.
"If you broke somethin', I can sure as hay take a look. That why you're out here?" the farmpony asked, still not entirely certain what Twilight was getting at.
Twilight shook her head.
"Oh, no. I'm just... well, it's about your cart, see. Your cart has a certain weight... that makes sense, right?"
Applejack narrowed her eyes. "Are we really doin' this?"
"Yes. So, the weight of the cart goes downwards, because of gravity. This force is balanced out by the ground, because... well, every force has an equal and opposite reaction."
Applejack sighed as she effortlessly pulled the cart up a hill, the harness pulling down on her neck a little. Apparently, Twilight wasn't going to stop talking any time soon.
"Uh huh. I think I heard that said someplace."
"Yes. Well, friction depends on the reaction force and whatever surface your cart is being dragged across."
"Uh huh."
"And your wheel is really really rusty, so let's say that there's a lot of friction between the ground and the cart."
"Uh huh. Are we negatin' air resistance?"
"Air... well, yes, we're not moving very fast. Anyway... well, what I'm trying to get at is the principle of superposition."
"I'll have you know I'm a very principled mare," Applejack told her, even though she knew that wasn't what Twilight was getting at. Some science-y thing, she imagined. Applejack had an appreciation for science, sure, but the only field she really felt she needed to know was economics. Specifically, how much money she would gain from this trip to town.
"The principle of superposition states that the total effect of loads on a body—"
"Hold up. Whose body are we talking 'bout?"
"Your cart."
"Just checkin'."
"The total effect of loading on a body can be represented as the sum of the individual loads affecting it." Twilight finished, a little peeved by the interruption.
Applejack forced back a chuckle. Playing around with Twilight was rather fun. She could see why Rainbow took such a strange amount of pleasure in it. It wasn't hard, either. Body? Loading? Why couldn't Twi just use ordinary words that didn't need any kind of context to understand? That's what happened when you had your head stuck in a book all day: you'd end up talking like one too.
"So, your cart is moving forwards, steadily. That means that there are no unbalanced forces, and that the sum of all these little forces equal zero."
"I still don't know why you're tellin' me this," Applejack said. Her patience was beginning to wear thin.
"Well, if one force decreases, but the sum of the forces remains the same, then that means that some other source is applying a force onto the body to keep it steady." Twilight explained. Applejack finally reached the top of the hill. Her cart, which was for some reason fascinating Twilight to no end, drifted lightly behind her, just as it had for a while now.
"Okay."
"I imagine the cart must feel lighter than it usually does, right?" Twilight told her, which caused Applejack to hesitate. It was true: the cart was being surprisingly easy to move today, despite it's age and weight.
"Uh... yeah, it is." she said, slowly and cautiously. Understanding felt like it was only a few seconds away. She narrowed her eyes at Twilight, who was grinning broadly.
Is she sayin' that there's an outside force makin' my cart feel lighter?
"Now that I've made my point, I'd like to say that you have some really patient friends, Applejack."
"Twi, if you're going to give me up, just do it already," came a tired voice from behind both mares. Applejack stopped in her tracks, the momentum of the cart pushing into her. With a clatter and a thud, the cart collided with the ground. The farmpony turned around.
Rainbow Dash was twirling her forelegs around, trying to get the blood flowing into her extremities once again. She smiled awkwardly at both Applejack and Twilight.
"Hi, guys."
She must have been carrying the cart from behind, Applejack realized. No wonder if felt so light going up the hill. She wondered why Rainbow hadn't said anything... and it must have been exhausting, too. The cart was heavy. To lift the whole thing up was a ridiculous idea, even for her.
"Have you been... helpin' me the whole trip?" Applejack asked. Rainbow shook her head.
"Oh, no. Just when your wheel broke on that stone."
Applejack looked over at her wheel: Rainbow was right. It had been broken. Several of the struts were snapped.
"I... uh, thank you." she mumbled.
"Hey, don't mention it. I didn't want you to feel you owed me or anything..." Rainbow told her, blushing slightly. "You know... just one friend helping another."
How very considerate, Applejack thought. Tartarus must have frozen over today for Rainbow to be so selfless. She turned towards Twilight, a little annoyed by the alicorn's secrecy. Still, it meant a lot that Rainbow was prepared to forego her usual bragging and boasting just to save Applejack from having a hard time.
"Well, I think thanks are in order anyhow," she said. To Twilight, she added, "why couldn't you have said that to start with?"
Twilight smiled innocently.
"I like to consider myself a teacher," she told them. "Also, It was fun watching Rainbow try to stay awake whilst listening to me ramble on about the laws of static mechanics."
Rainbow growled. "It wasn't fun for everypony."
Twilight chuckled. "Anyway, it's nice to see you two helping each other out. I know it made my day," she said, before stretching her wings, ready to fly. "I'd better get back to the library. I think this will make a nice little side note in the diary. See you guys later!"
As Twilight flew off towards Ponyville, both Applejack and Rainbow Dash turned back towards the broken cart.
"Looks like I'll have to drag that thing the rest of the way..." Applejack grumbled. Rainbow glanced back towards Twilight, almost a speck in the distance by this point.
"Y'know, I never realized how lazy Twilight was until just now," Rainbow added. "I mean, she's an alicorn now, and can lift crazy heavy stuff with her magic... but..."
Applejack frowned. "Y'know what? You're right. Remember the ursa minor?"
Rainbow nodded. They both looked at the cart, then back at the remainder of the distance to travel.
"Well... horsefeathers."
