In the Atwood machine discussed in Example 5.9 and shown in...

In the Atwood machine discussed in Example 5.9 and shown in Active Figure $5.14 \mathrm{a}, m_{1}=2.00 \mathrm{kg}$ and $m_{2}=$ 7.00 $\mathrm{kg}$ . The masses of the pulley and string are negligible by comparison. The pulley turns without friction, and the string does not stretch. The lighter object is released with a sharp push that sets it into motion at $v_{i}=2.40 \mathrm{m} / \mathrm{s}$ downward. (a) How far will $m_{1}$ descend below its initial level? (b) Find the velocity of $m_{1}$ after 1.80 $\mathrm{s}$ .

Key Concepts

Atwood Machine

An Atwood machine is a classic physics apparatus used to study motion under gravity. It consists of two masses connected by a string that passes over a frictionless, massless pulley. This setup allows for the analysis of acceleration and tension in a system where the only forces acting are gravity and the tension in the string, making it an ideal model for applying Newton’s laws of motion.

Newton's Second Law

Newton’s second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma), is fundamental in analyzing the motion in an Atwood machine. The law is used to set up equations for each mass to solve for the system’s acceleration and the tension in the string.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration and are essential when initial velocities are not zero. In problems like this, they help relate displacement, velocity, acceleration, and time, allowing one to determine quantities such as how far a mass moves or its speed after a given time.

Energy Conservation in Mechanics

The principle of energy conservation asserts that in the absence of non-conservative forces (such as friction), the total mechanical energy (the sum of kinetic and potential energy) of a system remains constant. Although sometimes a Newtonian force analysis is preferred, energy conservation can also be applied in problems like the Atwood machine to relate changes in potential energy to changes in kinetic energy.

Transcript

00:01 All right, let's solve problem 36 in chapter 5.

00:04 So we are given an atwood machine with two masses, m1 and m2.

00:18 And we're told that we, before we completely lay a loose, we give this a little push so it has a beginning speed of two meters per second downward.

00:29 And because m2 in this prom is greater than m1, what will happen is it will get slowed down by the mass of m2 and it'll reach some point before it then has to go back up again.

00:47 So we want to figure out what this distance is, and we also want to figure out what the speed of it after a certain period of time.

00:59 So first let's just solve the normal, the acceleration of this atwood machine.

01:08 So we know that if it's accelerating like this, for mass 2, it has the same acceleration for mass 1, right? because it's the same string as you think so.

01:22 If we look at mass 2, it has gravity and it has tension, right? and so if we're going to treat up, going up as positive, we get t minus m.

01:43 2g equals m2a right and we basically have the same picture so i won't bother drawing it twice for m1 except here downward has to be positive so we have to write m1 g minus t equals m1 right um and so what we can do is we can add these two equations together so we get m1 plus m2 g should be m1 minus m2 equals m1 plus m2 equals m1 plus m2 okay and therefore the acceleration is just m1 minus m2 over the sum of their masses times g and actually this is sort of an amusing question in that you can actually pretty much guess that this is the right answer right it almost has to be this answer.

03:08 You know that the acceleration of these objects has to be dependent on the difference between the two masses, right? because if they're the same mass, you would get just stand still.

03:18 It has to scale with some acceleration, and the only acceleration constant that's relevant is g.

03:27 But if you don't have this m1 over m2, you just have the top part and this g, it doesn't have the right units.

03:33 You have to divide by some mass, right? and it doesn't make sense to multiply by or to divide by only one of the masses.

03:39 It makes sense to use the sum.

03:42 And it definitely doesn't make sense to use the difference.

03:44 So you can pretty much like, you can guess this just by using dimensional analysis.

03:49 But in case, this is the answer.

03:51 It makes perfect sense.

03:52 So this is the acceleration we're going to get.

03:55 Now, note, because m1 here is lower than m2, you're going to get a negative acceleration when you plug in numbers.

04:02 And that's correct, right? because we're treating moving down for m1 is positive, but it's getting decelerated so the a should be negative this does have some consequences so let's just write for now a is negative still be a sort of a an important math trip up that some of you might run into if you're not too careful okay so we want to find the total range before or the range of it until it hits until it stops moving downward right so we know that the total speed at some time is just the initial speed plus acceleration times time...

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