Periodic assignment
.docx
School
Western University**We aren't endorsed by this school
Course
CALCULUS MCV4U
Subject
Mathematics
Date
Dec 18, 2024
Pages
4
Uploaded by MegaRamPerson962
Periodic assignment For my periodic assignment, I chose the clock because it is one of the most recognizable and fundamental examples of periodic motion in our everyday lives. The consistent, repeating movement of the hands around the clock face is a perfect demonstration of periodicity, as it follows a regular and predictable cycle. This constant repetition not only helps us measure time but also serves as a symbol of rhythm and order in our world. I found the clock intriguing because it seamlessly connects a mathematical concept, periodic motion, with something as essential as keeping track of time, making it both practical and symbolic.How is the clock hand periodic:The hands of a clock are periodic because they repeat their motion in a regular, predictable pattern over a fixed interval of time. Here's how each hand demonstrates periodicity:1. Second HandPeriod: 60 seconds (1 minute).The second hand completes one full rotation around the clock face every 60 seconds.The motion is periodic because the position of the second hand at any given second will repeat exactly every 60 seconds.2. Minute HandPeriod: 60 minutes (1 hour).The minute hand completes one full rotation around the clock faces every hour.Its position repeats in cycles of 60 minutes, making it periodic.3. Hour HandPeriod: 12 hours.The hour hand completes one full rotation around the clock face every 12 hours.Its motion is periodic as its position repeats in 12-hour cycles.Point of reference:My reference point is going to start when the hand is at 12. From here, I will use the ruler to measure the tip of the minute hands position to the new position. (Where t=0)
Hand displacement:Amplitude and vertical translation:Period of the function:Since our trigonometric functions is representing the distance of our minute hand terms of time passing in minutes, the functions period is 60. This is because it takes 60 minutes for the minute hand to come back to its original position.The starting value will be at t=0 to simplify the calculations to have no phase shift as it is the start of the hour. I will then use a ruler to measure the distance between the new distance from the minute hands original position.When the minute hand is at 45 minutes, displacement is 12cm. t=45, d=12cmWhen the minute hand isat 30 minutes, displacement is 24cm. t=30,d=24cmWhen the minute is at 0 minutes, displacement is 0cm. t=0, d=0cmWhen the minute hand is at 15 minutes, displacementis 12cm. t=15,
Create a function in the form of f(x)=asin(k(x-d)) =c and f(x)acos(x-d)) +cTo make our cos function into a sin function, all we need to do is change the starting point of t=0 to whatever d(t)=12. This because sin functions always start at the central axis. Thus, the equation is: Graphing:
Verifying the graph:To verify the graph, I chose to use Desmos after drawing it by hand. By doing so you can test each value that has to be in the graph and confirm if it works. Reflection:For me the most challenging part of this assignment was trying to model the movement of the clocks hand as a trigonometric function. While I can visualize how the hand moves as a function,translating that into an equation was a tough part. It’s crucial to ensure that all values are correct, as even a small mistake can throw off the entire calculation. The easiest part was understanding the clocks movement as a trigonometric function. Once I grasped that, I could solve for the variables, determine the equation, and graph the function. This assignment made me realize that periodic functions are more common than I initially thought and appear in many different contexts. Although there were many examples I could have used, I decided to focus on the simple idea of the clock hand. This showed me that many other types of functions can be modeled with trigonometric functions if the correct context and variables are chosen.