5BL Pre-Lab 4 - W24v2

Physics 5BL Pre-Lab 4: Oscillations Winter 2024, UCLA Department of Physics & Astronomy Directions: As you read through the pre-lab, follow along and complete the Google Slides pre-lab submission template to submit your responses to questions below on each slide as indicated. Background - Simple Oscillations: It is important to build fundamental physics experience with a mass-spring system or pendulum system in order to understand how physical properties influence the time-dependent (t) position x(t) and restoring forces F(t) oscillations in the system. Many aspects of living systems and modern technology rely on oscillations of material and/or particles, like how air molecule density oscillates to create sound waves. In sound, the air molecules’ high and low density fluctuations around atmospheric pressure are very similar to the scrunching and opening of a spring as an oscillating mass pushes and pulls on it. By understanding how the mass-spring system oscillates in this lab, greater appreciation and insight can be applied into the oscillations that enable sound, music, and hearing for next week’s lab. Specifically, whenever a system’s position is oscillating in time, we can describe these oscillations with information about the oscillation frequency and the oscillation amplitude. The amplitude describes the maximum displacement of the oscillation from the central (equilibrium) position and the frequency describes the number of oscillations (full cycles) per second (Hz) or it can be phrased as the number of radians ( full cycle) per second ( ⍵ ). 1 2π This means that there are two ways to express frequency: either in terms of revolutions per second or radians per second. The units of radians per second must be used when describing oscillations mathematically as a sine or cosine function, and the revolutions per second units are used when describing the number of times the entire cycle occurs every second. The main point of this lab is to understand and quantify how the oscillation frequency ⍵ is related to particular physical parameters of the system. Depending on the system, only certain parameters affect the oscillation frequency. Mathematically, one can derive the important physical parameters of an oscillating system by setting up the differential equation that relays Newton’s Second Law ( ) to the mathematical expression of net force in the system. 𝐹 = 𝑚𝑎 Since acceleration is the second derivative of position, a second order differential equation can be constructed and solved for the exact x(t) equation and the frequency of oscillation can be defined by the physical parameters in the equation (see the general solution in the supplementary materials). You do not need to know how to set up the mathematical equations for this lab, but you will practice drawing sine and cosine graphs to graphically see how position, velocity, are all related and completely defined by your setup initial conditions and parameters. By the end of the lab, you will be able to know what key information to ask your labmate about a trial setup, and then you will be able to draw the position, velocity, and acceleration graphs of that trial without actually having any measurement data. In the case of a mass-spring system, the oscillation frequency depends on the mass on the spring, and the stiffness of the spring. We will assume that the spring is so much lighter than the suspended mass that we can model the behavior using the assumption of a massless spring. In the case of the pendulum system, the oscillation frequency depends on the length of the string and the gravitational acceleration of the system. For the case of a spring, the frequency of oscillation does not depend on the amplitude of the oscillation. For the case of a pendulum, the amplitude does affect the frequency of oscillation for large deflection angles, but the 1

Physics 5BL Pre-Lab 4: Oscillations Winter 2024, UCLA Department of Physics & Astronomy amplitude does not affect the frequency of oscillation for small deflection angles. Your textbook has additional information about general oscillations, spring systems, pendulum systems, and the exact relationships between frequency and the physical parameters of the system. Check them out! Example simulations for exploration: Mass-spring system (select ‘Intro’) here Simple Pendulum system (select ‘Intro’) here Experiment Setup Information: You will create your own mass-spring system and pendulum system in the lab. There are two different spring options and two different mass options for your mass-spring system. The two springs are different lengths, and these different lengths result in unequal spring constants (you do not need to know why the length affects the spring stiffness, only that these springs have different spring constants and can be used to vary the parameters in your experiment). The masses are labeled, but you should weigh them to double-check anyway. In the case of the pendulum, different types of balls are attached to a string that you can set up to be the length of your choice. It is up to you to measure your string length and the mass of your ball (if you think that mass is something that you want to test). The pre-lab will help you prepare for what experimental setups you would like to test, to answer a particular question about angular frequency of a pendulum or a spring in different physical conditions. In 5AL, you used a camera and Tracker to measure oscillations (recall the Conservation of Energy lab with Potential Energy and Kinetic Energy calculations from position and velocity measurements). In this lab, you will experimentally measure the average frequency of the system by counting oscillations and measuring the time it takes for your system to perform a given number of oscillations (the number will be up to you, depending on what you think is reasonable). Then, with this experimental data, you will determine your experimental frequency, compare this experimental frequency to the prediction from your experimental setup measurements, and then plot the position vs. time of your system. You will do this for both the spring system and the pendulum system so that you will be able to compare/contrast the two systems at the end of your lab. Pre-Lab Assignment Pre-Lab Question 1: A hanging spring is stretched by 0.2 m when a mass of 0.5 kg is attached to it. The mass is then pulled down a further 0.1 m and released from rest. a. What is the angular frequency of the resulting oscillations? b. How many seconds does it take for the system to go through exactly 10 complete oscillations? Pre-Lab Question 2: Taking y = 0 as the initial equilibrium position of the mass in Question 1, and y positive upwards, write down an equation for the time dependence y(t) , giving the numerical values of any constants in the solution. 2