In a science museum, a 110 kg brass pendulum bob swings at the end of a 15.0-m-long wire. The pendulum is started at exactly 8:00 a.m. every morning by pulling it 1.5 m to the side and releasing it. Because of its compact shape and smooth surface, the pendulum's damping constant is only 0.010 kg/s. At exactly 12:00 noon, how many oscillations will the pendulum have completed and what is its amplitude?

Video transcript

Hello, fellow physicists today, we're gonna solve the falling practice prom 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 decorative pendulum on a wall clock is made of a 250 g, gold coated spherical mass hanging on a 35 centimeter long lace. The homeowner starts the pendulum at 4 p.m. by giving it a displacement of 3.0 centimeters on one side. The pendulum is nearly perfect with a damping constant of 2.5 multiplied by 10 to the power of negative 5 kg per second. Find the amplitude and the number of cycles the pendulum makes by 11 pm. So that's our end goals. We're trying to figure out the amplitude and the number of cycles that this pendulum will make by 11 pm. And that will be our final answer pair that we're ultimately trying to solve. For. Awesome. We're also given some multiple choice answers where N represents the number of cycles the pendulum makes by 11 pm and A of T represents the amplitude of the pendulum that it makes so the amplitude over time by 11 pm. So that's what A of T represents the function A T. So the amplitude is all in units of centimeters and the number of cycles are all in units of oscillations. So let's read them off to see what our final answer pair might be. A is 350 0.85. B is 21,000 and 0.85 C is 21,002 0.2. And finally D is 352.2. Awesome. So first off, let us assume that the decorative pendulum is a damped sci. So let us also note that by 11 pm, the pendulum will have oscillated for exactly or precisely 7.00 hours from four pm. So let's make a note of that. So from 11 pm to sorry, sorry, from four pm to 11 p.m. let's be exact here and not get confused here. So I, so from 4 p.m. So from 4 p.m. to 11 p.m. that means that it's been moving for 7.00 hours. So with that in mind, we now need to solve for the number of cycles that the pendulum will make. And in order to do that, we must first solve for the period of the pendulum. So let us recall that the equation for sulfur, the period is so capital t the period is equal to two multiplied by pi multiplied by the square root of L divided by G, where L denotes the length and G denotes gravity. So now we can plug in our known variables to solve for the period. So T is equal to two multiplied by pi multiplied by the square root of. So we're told that the length is 35 centimeters. So let's quickly convert 35 centimeters, 2 m. And when we do that, we will get 0.35 m divided by gravity, which the numerical value for gravity is 9.8 m per second squared. So when we plug that into our calculator, as you could see, our units of meters will cancel out leaving us with just seconds, which is perfect because we want our period to be in seconds. So when we plug that into our calculator, we will find that it's equal to 1.19 seconds when you round to two decimal places. So at this stage, we can now sol for the number of oscillations, which we're gonna denote as N by recalling and using the following equation that capital N, the number of oscillations is equal to the time divided by the period. So we can now plug in our known variables. So we know that the time that the pendulum's moving is for seven hours. So 7.00 hours. But or as you can see, we have a bit of a problem. If you haven't noticed it yet, we have our period, which is in units of seconds. So in order for our units to be consistent, we have to convert hours to seconds. So let's use some dimensional analysis to do that. So in seven hours, we know that there is 3600 seconds in one hour. So as you can see, the hour units will cancel out, leaving us with seconds, which is perfect, divided by the period, which we determine to be 1.19 seconds. Awesome. So our units of seconds will cancel out. And when we plug that into our calculator, we will find that it's equal to 21,176 which is approximately equal to 21,000 oscillations, which is our final answer. Hooray, we did it, we found our first answer. Now we need to solve for the amplitude. So to do that, we must recall and use the equation to solve for the amplitude which states that A of T is equal to capital, A multiplied by E to the power of negative B multiplied by T divided by two, multiplied by M where M is the mass of the pendulum. B is the damping constant T is the time of oscillation. And capital A denotes the initial displacement of the pendulum. So with that in mind, we can now plug in all of our known variables to solve for our final answer the amplitude. So let's do that. So a of T is equal to A. So the value for A is 0.030 m because we're told in the problem itself that the initial displacement of the pendulum is three centimeters, but we need to convert three centimeters to meters and then we need to multiply it by E to the power of negative. And we know that the value for B is given to us as 2.5 multiplied by 10 to the power of negative five. And its units are kilograms per second. And this is multiplied by T which when you multiply the value for T which this is the 7.00 hours multiplied by 3600 seconds, we will find that that's equal to 25,200 seconds. And this is all divided by two multiplied by 0.25 kilograms. Because we're told that in the prom itself, we're told that it has a mass of 250 g. So we need to convert grams to kilograms in order to get our units to match up. So everything cancels out the way it should to get our units and the answers that we want to get the units that we want, which in this case, we're gonna get units of meters and then want to convert meters to centimeters to match our multiple choice answers. So when we plug that into our calculator, we will find that aft, the amplitude is equal to 8.5 multiplied by 10 to the power of negative 3 m like I was discussing, but we need to multiply by 100 in order to convert meters to centimeters. And when we do that, we will get zero point 85. And note that when we have it originally in meters, this is when we round to one decimal place. So our final answer has to be 0.85 centimeters and we did it we solved for this problem. So looking at our multiple choice answers, the correct answer has to be the letter B which states that N is equal to 2 21,000 oscillations and the amplitude A of T is equal to 0.85 centimeters. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.