A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7
shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Question

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7

Graph showing the position of a 2.40-kg ball oscillating on a spring over time.

shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Graph depicting the oscillation of a ball on a spring, showing position versus time.

Accepted Answer

Everyone in this problem. A spring of unknown force constant has a 3.2 kg block fixed to one of its ends. The block position as a function of time is shown in the graph below and were asked to determine the amplitude of the oscillation in the springs force constant. All right. So, if you look at this graph were given the position X and centimeters. And the time t in seconds. Now, the first thing we want to find is the amplitude. Okay, now the amplitude A is going to be the maximum displacement from X equals zero. Okay, So we want to take that maximum height cause of our negative and we see that we get down to negative six centimeters and then we have a maximum here of six centimeters. And so our amplitude A is going to be six centimeters. Okay, it's that max displacement from X equals zero. Okay. Alright, so we found our amplitude a. It's six cm now. We need to go ahead and find the springs force constant. Okay. All right, well, let's think about the force cost. Okay, we know that omega. The angular frequency is equal to two pi over the period two, which is equal to the square root of K. Overall. Alright, we're trying to find K. This force constant. We know the mass. M. Okay, so if we can find the period T from our graph that we can solve for K. Okay. All right, well, what we'll notice we start at zero here. Can we go down and back up and right here at 0.3 seconds. Okay, we're back at zero and now we're going upwards. Okay? So if we continue this graph this here with the dotted line would be a full period. Okay? And this point here is halfway. Okay? So that tells us that half of the period T over two Is going to be equal to 0.3 seconds because we've gone from zero seconds to 0.3 seconds. So it's taken a 0.3 seconds. So go halfway. Okay, so half of our period is 0.3 seconds. Which tells us that T is equal to 0.6 seconds. All right, So, we know our period T. We know our mass M. We can use this equation to solve for K. So we have two pi over the period 0. seconds is equal to the square root of K. What? We're trying to find that force constant. The spring force constant Divided by the mass of 3.2 kg. Okay. All right, let's give ourselves some more room. Now, if we square both sides and we get two pi over 0.6 seconds squared is equal to K over 3.2 kg. And if we multiply by 3.2 kg, we get that K is equal to two pi over 0.6 seconds squared times 3. kg. Okay? And this is gonna give us a K value .92 Newtons. Her meter. Okay. Alright. So we found our amplitude A. We found our spring force constant K. If we look at our answer choices, we have an amplitude of six centimeters K. So we're looking at either a B or C. And then we have found a. K value of 350.92 newton meters. We can approximate to the nearest newton meter. 351 newtons per meter. So we get the answer is C. Thanks everyone for watching. I hope this video helped see you in the next one.

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7 shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?

Verified Solution

A 2.40-kg ball is attached to an unknown spring and allowed to oscillate. Figure E14.7
shows a graph of the ball's position x as a function of time t. What are the oscillation's (a) period, (b) frequency, (c) angular frequency, and (d) amplitude? (e) What is the force constant of the spring?