Good Vibes Introduction to Oscillations Description Several conceptual and qualitative questions related to main characteristics of simple harmonic motion amplitude displacement period frequency angular frequency etc Both graphs and equations are used Learning Goal To learn the basic terminology and relationships among the main characteristics of simple harmonic motion Motion that repeats itself over and over is called periodic motion There are many examples of periodic motion the earth revolving around the sun an elastic ball bouncing up and down or a block attached to a spring oscillating back and forth The last example differs from the first two in that it represents a special kind of periodic motion called simple harmonic motion The conditions that lead to simple harmonic motion are as follows There must be a position of stable equilibrium There must be a restoring force acting on the oscillating object The direction of this force must always point toward the equilibrium and its magnitude must be directly proportional to the magnitude of the object s displacement from its equilibrium position Mathematically the restoring force is given by where is the displacement from equilibrium and is a constant that depends on the properties of the oscillating system The resistive forces in the system must be reasonably small In this problem we will introduce some of the basic quantities that describe oscillations and the relationships among them Consider a block of mass attached to a spring with force constant as shown in the The spring can be either figure stretched or compressed The block slides on a frictionless horizontal surface as shown When the spring is relaxed the block is located at to the right a distance oscillations and then released If the block is pulled will be the amplitude of the resulting Assume that the mechanical energy of the block spring system remains unchanged in the subsequent motion of the block Part A After the block is released from ANSWER remain at rest it will move to the left until it reaches equilibrium and stop there move to the left until it reaches and stop there move to the left until it reaches to the right and then begin to move As the block begins its motion to the left it accelerates Although the restoring force decreases as the block approaches equilibrium it still pulls the block to the left so by the time the equilibrium position is reached the block has gained some speed It will therefore pass the equilibrium position and keep moving compressing the spring The spring will now be pushing the block to the right and the block will slow down temporarily coming to rest at After is reached the block will begin its motion to the right pushed by the spring The block will pass the equilibrium position and continue until it reaches completing one cycle of motion The motion will then repeat if as we ve assumed there is no friction the motion will repeat indefinitely The time it takes the block to complete one cycle is called the period Usually the period is denoted and is measured in seconds The frequency denoted is the number of cycles that are completed per unit of time In SI units is measured in inverse seconds or hertz Part B If the period is doubled the frequency is ANSWER unchanged doubled halved Part C An oscillating object takes 0 10 to complete one cycle that is its period is 0 10 What is its frequency Express your answer in hertz ANSWER Part D If the frequency is 40 what is the period Express your answer in seconds ANSWER The following questions refer to the figure that graphically depicts the oscillations of the block on the spring Note that the vertical axis represents the x coordinate of the oscillating object and the horizontal axis represents time Part E Which points on the x axis are located a distance ANSWER R only from the equilibrium position Q only both R and Q Part F Suppose that the period is Which of the following points on the t axis are separated by the time interval ANSWER K and L K and M K and P L and N M and P K and P are separated by phase interval of 2 or the block gets back to the same point Now assume that the x coordinate of point R is 0 12 is 0 0050 Part G and the t coordinate of point K What is the period Hint G 1 How to approach the problem In moving from the point to the point K what fraction of a full wavelength is covered Call that fraction Then you can set Dividing by the fraction will give the period Express your answer in seconds ANSWER Part H How much time does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement Express your answer in seconds ANSWER The block travels only half the period Part I What distance does the object cover during one period of oscillation Express your answer in meters ANSWER 4 times of the amplitude Part J What distance does the object cover between the moments labeled K and N on the graph Express your answer in meters ANSWER 3 times of the amplitude Energy of a Spring Description Short quantitative problem relating the total potential and kinetic energies of a mass that is attached to a spring and undergoing simple harmonic motion This problem is based on Young Geller Quantitative Analysis 11 1 An object of mass attached to a spring of force constant oscillates with simple harmonic motion The maximum displacement from equilibrium is mechanical energy of the system is Part A and the total What is the system s potential energy when its kinetic energy is equal to Hint A 1 How to approach the problem Since the sum of kinetic and potential energies of the system is equal to the system s total energy if you know the fraction of total energy corresponding to kinetic energy you can calculate how much energy is potential energy Moreover using conservation of energy you can calculate the system s total energy in terms of the given quantities and At this point you simply need to combine those results to find the potential energy of the system in terms of and Part A 2 Find the fraction of total energy that is potential energy When the kinetic energy of the system is equal to what fraction of the total energy is potential energy Hint A 2 a Conservation of mechanical energy In a system where no forces other than gravitational and elastic forces do work the sum of kinetic energy and potential energy energy of the system given by Express your answer numerically ANSWER is conserved That is the total is constant Part A 3 Find the total energy of
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