A block of mass 0.300 kg is attached to a spring. At x = 0.240 m, its acceleration is a x = -12.0 m/s 2 and its velocity is v x =4.00 m/s. What are the system's (a) force constant k and (b) amplitude of motion?

Hey, guys. Hopefully, you didn't have too much trouble with this. So we've got a block of mass 0.3 kilograms and it's attached to a spring. At a specific point, we're told that the acceleration is something and that the velocity is also 4 meters per second. And we're supposed to figure out what the force constant is. So let's take a look at all of our variables. We're trying to figure out what k is, and I'm just going to write out all of my variables. I know that the mass is equal to 0.3. I'm told that x is equal to 0.24. And at that specific point, a of x so let me write that in actual, like, notation. A of x is equal to negative 12, that's in meters per second squared, and v at that particular point is equal to 4, and that's in meters per second. Okay. And now we're supposed to figure out what k is. So let's just go to our equations and figure out which one we're using. Well, let's just start from the top. Okay. So whenever we use these equations, we're given something about the force or the acceleration, and we are actually. So if you look at this equation here, am I gonna use the max or am I gonna use the a equation? Well, let's take a look. If this equation, if I'm using this equation, I know what the acceleration is at a specific point. I'm trying to figure out what the spring constant k is. I'm given the x, and I also have what the mass is. So I can use that equation. So let's move over here. So I'm gonna use this equation. So a of x is equal to negative k over m times x. But what are we told? We're told that a of x is equal to negative 12. So that means that So both of these things, a of x is equal to negative 12. I could just set these 2 equal to each other. Right? I could just set this guy and this guy equal to each other. So I've got negative k over m times x is equal to negative 12. The good thing is that the negative signs will go away. Now, I'm just gonna start plugging in all my numbers. Right? So I'm looking for k. Now m is equal to 0.3, and then the x distance is 0.24, and that equals 12. So if you just divide and multiply all this stuff over, you're gonna get 12 times 0.3 divided by 0.2 4. And what you're gonna get is exactly 15 newtons per meter. That is our spring constant. So let's move on. So we got part b and we're supposed to ask now what the we're supposed to find out what the amplitude of motion is. So we're supposed to figure out this guy. What are the amplitude of this mass spring system? Okay. So now we're looking for amplitude. Let's check it out. So we start from the top. Right? We're not given any maximum acceleration, so we got a bunch of stuff that involves a here. So we're not given any maximum force or maximum acceleration. We can't use that. We also aren't given a specific time to evaluate any of our variables for, so we can't use any of those cosine sine ones. We also just we don't know anything about the v max, a max, anything like that, so we can't use those. So whenever this happens, use your energy conservation equations. So let's take a look. We've got 2 of them. I've got an amplitude in this one and an amplitude in this one. So let's take a look. What do I know? I know what the k constant is. I just figured that out. I don't know what the v max squared is, so I I can't use that one. And if you look here, I've got a k that I know. Right? And I've got the x that I know. So I have x at a specific position. And I also have m and velocity at that position. So I have all of those things. So what I'm gonna use is I'm gonna use this equation right here. The ka squared equals 1 half kxp squared plus 1 half mvp squared. That's a mouthful. Okay. So I've got 1 half k a squared, and then one half k x squared 1 half mv squared. Okay. So one thing is that I've got halfs in all of these terms, so those will end up just canceling out. So whenever you have, like, the same fraction or something like that in all the terms, they just canceled out. Okay. And so I'm looking for the amplitude. So let me go ahead and highlight that. I've got the amplitude. So what I can just do is just divide all of this. I could just divide the k over to the other side and just start plugging in numbers. So a squared is gonna be well, let's see. K is equal to 15. The x that I'm evaluating this at is 0.24. That's gonna be squared. Then I've got the mass, which is 0.3 times the velocity, and the velocity of that specific point is 4 meters per second. I'm gonna square that. Then I'm gonna divide the whole entire thing by the k that I'm dividing over to the other side. So that's 15. And so what happens is if you plug all of that stuff and then you take the square root of that, the amplitude is gonna be 0 0.61 meters. And that is the answer. Alright, guys. Let me know if you guys have any questions with this. I'll be happy to help you out. Let's keep going.

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17. Periodic Motion

Energy in Simple Harmonic Motion

Physics

17. Periodic Motion

Energy in Simple Harmonic Motion

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Multiple Choice

A block of mass 0.300 kg is attached to a spring. At x = 0.240 m, its acceleration is a_x= -12.0 m/s² and its velocity is v_x=4.00 m/s. What are the system's (a) force constant k and (b) amplitude of motion?

9 N/m; .35 m

15 N/m; .61 m

15 N/m; 1.22 m

9 N/m; .11 m

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Verified step by step guidance

Start by using Hooke's Law to find the force constant k. Hooke's Law is given by F = -kx, where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.

Use Newton's second law, F = ma, to express the force in terms of mass and acceleration. Here, F = m * ax, where m is the mass of the block and ax is the acceleration. Substitute this into Hooke's Law to get m * ax = -k * x.

Rearrange the equation to solve for the spring constant k: k = -(m * ax) / x. Substitute the given values: m = 0.300 kg, ax = -12.0 m/s², and x = 0.240 m.

To find the amplitude of motion, use the energy conservation principle. The total mechanical energy in a spring-mass system is the sum of kinetic and potential energy: E = (1/2)mv² + (1/2)kx². At maximum displacement (amplitude), all energy is potential: E = (1/2)kA², where A is the amplitude.

Set the total energy at x = 0.240 m equal to the potential energy at maximum amplitude: (1/2)mv² + (1/2)kx² = (1/2)kA². Solve for A by substituting the known values for m, v, k, and x, and then rearrange the equation to find A.

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