welcome to the physics classrooms video tutorial on vibration and waves the topic of this video is periodic motion and we want to know what traits characterize an object that undergoes periodic motion and how do the quantities period frequency and amplitude describe such an object I'm Mr H let's get started this Mass on the end of a spring is vibrating back and forth about a fixed position when I think about how to describe it two words come to mind first of all its motion is repeating it occurs over and over again cycle after cycle and second its motion is regular it occurs in the same amount of time cycle after cycle any object whose motion can be described by these two words repeating and regular is an object that's undergoing periodic Motion in the phrase periodic motion you hear the word period a physics term which means the amount of time that it takes to complete one full cycle of vibration any object that's undergoing periodic motion always takes the same amount of time to complete each consecutive cycle its period is constant for this Mass on the spring if it takes 2.5 seconds to complete its first cycle then it takes 2.5 seconds to complete its fifth cycle in its nice ninth cycle and so on a mass on the end of a spring is not the only example of periodic motion the back and forth motion of a bob hanging from a string a pendulum is also undergoing periodic motion the rotation of the earth about its axis is also an example of periodic motion it always takes 24 hours for the Earth to complete one full rotational cycle you can set your clock to that and the orbit of the earth about the sun is another example of an object that is in periodic motion it always takes 365.25 days for the earth to orbit the Sun and you can set your calendar by that one if we were to place a motion detector below a vibrating Mass on the end of a spring in order to detect the position of that mass as a function of time we would receive a plot that looks something like this this plot has the shape of a sine wave if you've ever plotted y equals sine of X on your graphing calculator you would receive a plot that looks something like this in this plot we recognize that that mass is vibrating back and forth about a fixed position its nearest position to the motion detector is 20 centimeters in its furthest distance from the motion detector is 120 centimeters it's vibrating back and forth from 20 centimeters away to 120 centimeters away about the fixed position of 70 centimeters this 70 centimeters position is known as the resting position it's a position that this Mass would assume if it were not vibrating up and down the position of this Mass varies as a function of the sign of the time any two quantity in which one quantity varies as a function of the sine of the other quantity are said to have a sinusoidal relationship earlier I said there are two words that describe an object in periodic motion regular in repeating the object vibrates back and forth repeatedly over the course of time in a regular manner we can look at this position time graph representing the motion of the vibrating Mass on the end of the spring and we notice that there's a shape on the graph that repeats itself if you start at time equals zero in 70 centimeters position you notice that the graph goes from 70 up to 120 back down to 70 then it goes down to 20 and back up to 70 again that shape up down down up repeats itself over and over and over again about this position time graph so is it re is it a regular repetition that is does it always take the same amount of time in order to test I could take that section of the graph and I could just pull it out and just move it over and then move it over again and then move it over again cycle after cycle after cycle to see if it matches up and what we notice is that it does match up it always takes the same amount of time for that graph to go up down down up that period is a constant period when we if we were to say that one unit along that time axis equals a second one square equals second then what we notice is that the first cycle goes from 0 to 6.3 seconds the second cycle goes from 6.3 to 12.6 seconds the third cycle goes from 12.6 seconds to 18.9 seconds and so on and so forth what we notice is that the period is always 6.3 seconds the period of an object that's undergoing periodic motion is always constant a vibrating Mass on the end of a spring experiences damping the result of the interaction of the mass spring system with the surroundings friction are resistance and other forces cause the energy of the system to gradually dissipate to the surroundings over the course of time and we would observe that the amplitude of that vibrating Mass would gradually diminish over the course of time instead of vibrating from 120 centimeters at the high position to 20 centimeters at the low position back and forth over the course of time those two extreme positions would become closer to the resting position but the period of vibration remains constant consistent with an object in periodic motion a plot of position versus time would look something like this you'll notice that the amplitude of vibration diminishes over the course of time when I say amplitude I mean the amount of displacement of the mass from that resting position denoted by the blue Dodge dashed line on this graph now as you see a damped Mass on a spring you don't want to say that it's slowing down because slowing down doesn't describe that Mass on the spring slowing down refers to the speed of an object and as we'll discuss on the next slide over the course of one cycle the mass on the spring is both speeding up and slowing down so if slowing down does it describe the mass on the spring how would you describe it well you would first of all use the term period is constant the period is always the same if you look at this particular graph and you look for the repeating part portion of the graph you'll notice it takes two seconds for every cycle you can just look and find that repeating section go over to the next section to the next section cycle after cycle and it's always taking two seconds so the period is constant it's not slowing down now what you do want to say is you want to say the amplitude is decreasing or diminishing over the course of time and that's because of damping for a vibrating Mass on a spring the position varies sinusoidally as a function of the time but position time is not the only sinusoidal relationship so is the velocity time relationship when we speak of the Velocity in physics class we're referring to the speed with the direction it has a number which represents how fast the object is going and it has a plus minus sign which tells us which direction the object is going so when we say the velocity is positive 1 meters per second we mean the speed is 1 meters per second in the direction it's moving is in the positive direction now here's the relationship between velocity time for a vibrating Mass on a spring you'll notice the sinusoidal look of this particular graph if we go through the graph starting at a time of zero seconds from the zero seconds to the end of the graph we would notice these types of changes first in the first approximately 1.6 seconds this mass is slowing down from one meter per sec in to zero meters per second and at 1.6 seconds it changes its direction and starts moving in the negative Direction so we see negative velocities plotted and it begins to speed up from zero meters per second to one meter per second in the negative Direction until about 3.2 3.3 seconds at that point in time the mass begins to slow back down from negative one meter per second to zero meters per second in the next 1.6 seconds at approximately 4.9 seconds the Mass is at moving zero meters per second and it changes direction again as the line on the graph crosses over into the positive region denoting that the mass is now moving in the positive direction and it speeds up from about 4.9 seconds to about 6.3 seconds to a speed of 1 meters per second in the positive direction now this happens over and over and over again repeatedly over the course of time the mass slows down changes Direction speeds up slows down changes Direction speeds up slows down changes Direction on and on and on again now if you look for the repeating cycle in this graph it starts high and then it goes down to the lowest point and then back up to the highest point and that repeats itself over and over and over again over the course of time and if you presume that this little square on the time axis represents one second it happens over and over again every 6.3 seconds a sign that the period is constant cycle after cycle after cycle I've been using the term period to describe an object undergoing periodic motion the term period answers the question how much time does it take to complete a cycle another term we could use is the term frequency which describes the number of complete vibrational cycles per unit of time if period answers the question how much time then frequency answers the question how often let's look at the formulas for a period in frequency the formula for period is period or t equal the time in seconds divided by the number of complete Cycles the equation for frequency is frequency equal or F equal the number of Cycles divided by the time you'll notice that these two expressions are reciprocals of one another now let's do an example suppose we have an object that completes 60 Cycles in 10 seconds perhaps it's like needle and push-ups in this situation if we were to calculate the period we would take the time 15 seconds and divide by the number of Cycles 60 cycles and we'd get a result of 0.25 seconds per cycle and if we were to calculate the frequency we would take the number of Cycles 60 and divide it by the time 15 seconds and we would get 4.0 cycles per second now what do these numbers tell us or what do they mean well first the period number of 0.25 seconds answers the question how much time does it take Mr H to do a push-up and the frequency number of 4.0 cycles per second answer the question how often does he complete a push-up and the answer would be four times per second the second thing that these numbers mean is it's at this time in every video that I like to help you out with an action plan a series of next steps for making the learning stick but before I help you out can you help us out by giving the video a like subscribing to the channel leaving a question or comment in the comment section below now for your action plan here are three resources that you'll find on our website any and one of which would be good next stops I've left links to each of these three in the description section of this video you have a simulation page on a vibrating Mass on a spring you have a tutorial page on periodic motion and you have a calculator pad problem set on the topic of frequency and period whatever you do I wish you the best of luck I'm Mr H and I thank you for watching