ShmoopTube

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Here's your shmoop du jour, brought to you by the falling apple, sponsored by gravity.

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Your tree house has a sweet system for bringing food up to you without making you go down the rope ladder.

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Two baskets are connected by a light string over a massless pulley.

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One rests at the bottom of the tree, and the other is at the level of the tree house.

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When your mom comes by and puts your box of Lunchables in the lower basket, you add rocks

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to the upper basket. The heavier basket falls, and the lighter basket rises.

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Lunchables are served!  If we assume that the pulley is frictionless,

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the basket of highly processed snack food has a mass of 2 kilograms, and the baskets

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of rocks has a mass of 6 kilograms... ...with what acceleration does your lunch arrive?

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And here are the possible answers...

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Before we even do any calculations, we can eliminate every choice except for A or B.

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Objects near the surface of the earth accelerate at 9.8 meters per second squared, which

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can round to about 10.

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However, since we have two baskets, accelerated

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by gravity working against each other,

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we know that the net acceleration MUST be less than 10.

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We don't know exactly how much less, but we're left with A and B.

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This is a pretty complex system, and finding

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acceleration won't be as simple as eating that delicious lunch dear old Mom sent you.

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But a little complexity never stopped us... To find acceleration, we need to look at each

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basket and the forces acting on that basket.

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To start, we'll look at the one with the food in it.

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The basket has two forces on it: Gravity and tension.

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Gravity points down, as always, and tension points up.

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Now, when we're solving a physics problem, we can't go wrong with F = m times a.

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We know that the net force on this object is equal to its mass times its acceleration,

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which we're trying to find. The net force is also equal to the sum of

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the forces acting on the basket.

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Tension and gravity are working in opposite directions,

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so one must be negative.

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Since our lunch is accelerating up... which hopefully won't happen after we eat it...

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...we will call that one positive, and down will be negative.

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Our net force is then also equal to T minus m times g.

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The second basket, the one with the rocks,

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is in a similar situation. We use F = m times a to find that the net

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force on the basket with the rocks is also equal to its mass times acceleration, and

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the sum of the forces acting on it. But wait! We almost made a terrible mistake.

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For the basket with the lunch, we made acceleration upwards positive, so we have to keep it consistent...

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The basket with the rocks is accelerating downwards, so we have to make acceleration

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negative. Our equation then looks like this: We now have two equations that we know to

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be true about this system: We know the mass of the two baskets, and the

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force of gravity... which leaves only acceleration, which we're trying to find... and tension.

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We know that acceleration and tension must be the same for the two baskets because they're

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part of the same system.

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This leaves us with a system of two equations for us to solve.

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To begin, we isolate T for both equations.

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We add m times g to both sides of both equations, and we get the following:

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Then, we set the two equations equal to each other, so we get that the mass of lunch times

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acceleration... plus the mass of lunch times the acceleration due to gravity... is equal

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to the negative mass of the rocks times acceleration... plus the mass of the rocks times gravity.

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Plugging in values, we get 2a plus 2 times 10 is equal to negative 6 times a plus 6 times 10.

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We add 6a to both sides and subtract 20 from both sides to get 8a is equal to 40.

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Finally, we divide both sides by 8 to get

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the acceleration is equal to 5 meters per second squared.

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So the basket accelerates at 5 meters per second squared...

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...which is answer B. As in, "Barely edible."

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