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A 0.500-kg glider, attached to the end of an ideal spring with force constant $k =$ 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute (a) the maximum speed of the glider; (b) the speed of the glider when it is at $x = -$0.015 m; (c) the magnitude of the maximum acceleration of the glider; (d) the acceleration of the glider at $x = -$0.015 m; (e) the total mechanical energy of the glider at any point in its motion.

A. $1.20 \mathrm{m} / \mathrm{s}$

B. $1.11 \mathrm{m} / \mathrm{s}$

C. $36 \mathrm{m} / \mathrm{s}^{2}$

D. $+13.5 \mathrm{m} / \mathrm{s}^{2}$

E. $0.360 \mathrm{J}$

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{'transcript': "let us write down two important equations that will help us solve the problem. We know that the total energy of the simple harmonic oscillator is equal to its kinetic energy. That's its potential. Oh, you know his kinetic energy. It's equal to 1/2 empty squared, and its potential is equal to 1/2 K. Hey, squirt. Okay, Expert Sorry about the K Express. They also know that the total energy of simple harmonic oscillator is equal to 1/2 K a squared, whereas the impotent also noon Second law for simple harmonic oscillator gives us that negative K X, which is the restoring linear force Bye hopes Law is equal to mass times acceleration. And so this top equation relates ex with the with the amplitude and the bottom equation relates acts with the acceleration. And so in party, the velocity is a maximum whenever x zero. So we're gonna plug in X is equal to zero in this top equation and sulfur B, and that will be our maximum velocity. Plugging that end gives 1/2 M v. Mac's squared equals 1/2 k squared. Everything in here is known except for the velocity, which is what we're solving for, and Sophie Max is equal to a square root Kayla Graham. And then we can just plug in the constant to give us. And this is equal to 1.2 meters per second. And that's answer to party for part B. We want to sell for the loss of you. One of her ex is equal to negative 10.15 meters. And so for this we're going to use this equation again. Except we're not going to plug in X equals zero. We're gonna plug in that X is equal this and we're going to solve her V again. When we do that, we get thie is equal to plus or minus square root of K over him, Times square root of a squared, minus X squared. When we plug in the values that the problem gives us, we get plus or minus 1.11 years for second. The reason we have plus or minus is because at one point the mass is going to be moving to the right at 1.11 years for second, and then it will slow down and we'll start exploring this way and then when it passes this point, it's going to move the same speed, but in the opposite direction. That's what the minuses indicated here when reality is talking about the speed. So we just say that the speed is 1.11 meters per second. Since he doesn't care about the direction we can read the mind Son and Parsi. We need the max acceleration, and we know the max acceleration occurs wherever the object is not moving and X is equal to. Plus, you might say this is a standard fact of simple harmonic motion is that X is equal to plus on my essay. The acceleration is the max and so going back to the first page here we can use this equation, except we're gonna plug in that X is equal day and sulfur What little a's So a max is equal to K over him, and now we can plug in the values that throne gives us. And when we do that, we have 36 meters per second squared, and that's the answer to Percy Parte. De wants the acceleration whenever X is equal to negative 0.15 And so for this we can just take the same equation that we just used. Except instead of replacing X with a, we're going to a place of Native 0.15 Other than that, it's exactly the same. And so when he put that and you get 13.5 meters per second squared party wants the total energy in this. And we know based on here that he is equal to 1/2 k examples of squared. So just rewrite that. It's 1/2 K a squared and everything here we know so we can split it, and that's what we get, and that's that."}