(II) At t = 0, an 885-g mass at rest on the end of a hori...

Question

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine

(c) the maximum acceleration,

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A cart of mass 1.5 kg is initially at rest on a smooth horizontal surface. One end of the cart is attached to a horizontal spring with a spring constant of 230 newtons per meter. Suddenly it is kicked by a person giving it an initial speed of 3.0 m per second. What is the maximum acceleration experienced by the cart? So that's our final answer that we're ultimately trying to solve for our end goal. So ultimately, we're trying to figure out what the maximum acceleration of the cart is and that's gonna be our final answer. So looking at our multiple choice answers, we can see that they're all in units of meters per second squared, which is the standard units for acceleration. So let's read them off to see what our final answer might be. A is 3.0. B is 9.0 C is 37 and D is 84. So first off, let us note that the initial speed of the car is the maximum speed. Let us also note that at this speed, we will have the maximum displacement of the spring take place. So at the maximum displacement, all of the kinetic energy of the car will be converted into potential energy. Now, at this point, we need to recall and apply the conservation of mechanical energy which states that the kinetic energy ke is equal to the potential energy pe. So we can also write, we can also write this out. So we know that the kinetic energy equation is one half multiplied by the mass multiplied by the maximum velocity squared, which is equal to the potential energy, which is one half multiplied by the spring constant multiplied by the area squared. Awesome. Where once again, M is the mass of the cart in this particular case. And that's 1.5 kg. The maximum speed is also equal to the initial speed which is equal to 3.0 m per second. And our spring constant K is equal to 230 newtons per meter. So now in order to solve for our maximum acceleration, which is our final answer we're ultimately trying to solve for, we must first rearrange our mechanical energy equation to isolate and solve for capital A means we need to use that value to solve for our maximum acceleration. So when we rearrange our mechanical energy equation to sulfur. The area, we will find that a, our area is equal to the square root of the maximum velocity squared multiply by the mass all divided by the spring constant K. So when we plug in our known variables and solve for the numerical value of A or area, we know that V max is equal to 3.0 m per second. And this value is squared multiplied by 1.5 kg because that's the mass of our cart all divided by our spring constant, which is 230. And don't forget that the units are gonna be newtons per meter. Awesome. So let me plug that into our calculator. We will find that a the area is equal to 0.2423 meters will be round to four decimal places. So now we can officially solve for the maximum acceleration by recalling and using the following equation that a max and we're gonna use lowercase A max to represent the maximum acceleration which is equal to the spring constant multiplied by the area all divided by the mass. So plugging in our known variables, we know our spring constant is 230 newtons per meter multiplied by 0.2423 m for our area all divided by our mass, which is 1.5 kg. So when we plug that into our calculator, we will find that a max is equal to 37.15 m per second squared, which rounding to the nearest whole number or to two significant figures. Because as we can see all of our multiple choice answers are rounded to two significant figures. We will get 37 m per second squared. And that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter C 37 m per second squared. Thank you so much for watching. Hopefully that helped and I can't wait to see in the next video. Bye.

Struggling with Physics?

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine
(c) the maximum acceleration,

Watch next

Intro to Simple Harmonic Motion (Horizontal Springs) practice set

Struggling with Physics?

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine(c) the maximum acceleration,

Watch next

Intro to Simple Harmonic Motion (Horizontal Springs) practice set

(II) At t = 0, an 885-g mass at rest on the end of a horizontal spring (k = 184 N/m) is struck by a hammer which gives it an initial speed of 2.12 m/s. Determine
(c) the maximum acceleration,