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AP® Physics B: Newtonian Mechanics Drill 1, Problem 3. With what acceleration does lunch arrive?

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Transcript

00:03

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

00:09

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

00:16

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

00:24

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

00:28

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

00:33

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

00:37

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

00:42

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

00:48

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

00:55

And here are the possible answers...

01:00

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

01:06

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

01:12

can round to about 10.

01:15

However, since we have two baskets, accelerated

01:18

by gravity working against each other,

01:20

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

01:31

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

01:36

This is a pretty complex system, and finding

01:37

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

01:42

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

01:47

basket and the forces acting on that basket.

01:51

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

01:54

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

01:59

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

02:04

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

02:09

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

02:14

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

02:18

the forces acting on the basket.

02:25

Tension and gravity are working in opposite directions,

02:28

so one must be negative.

02:30

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

02:35

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

02:40

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

02:48

The second basket, the one with the rocks,

02:50

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

02:55

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

03:00

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

03:06

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

03:12

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

03:17

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

03:23

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

03:27

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

03:33

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

03:37

part of the same system.

03:39

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

03:43

To begin, we isolate T for both equations.

03:46

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

03:57

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

04:01

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

04:06

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

04:12

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

04:23

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

04:29

Finally, we divide both sides by 8 to get

04:31

the acceleration is equal to 5 meters per second squared.

04:35

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

04:38

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

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