Oscillations AP Physics C Oscillations and Simple Harmonic Motion 1
Equilibrium and Oscillations A marble that is free to roll inside a spherical bowl has an equilibrium position at the bottom of the bowl where it will rest with no net force on it. If pushed away from equilibrium, the marble s weight leads to a net force toward the equilibrium position. This force is the restoring force. When the marble is released from the side, it does not stop at the bottom of the bowl; it rolls up and down each side of the bowl, moving through the equilibrium position. This repetitive motion is called oscillation. Any oscillation is characterized by a period and frequency. Frequency and Period For an oscillation, the time to complete one full cycle is called the period (T) of the oscillation. The number of cycles per second is called the frequency (f ) of the oscillation. The units of frequency are hertz (Hz), or 1 s 1. 2
Example A mass oscillates on a horizontal spring with period T 2.0 s. What is the frequency? If the mass is pulled to the right and then released, how long will it take for the mass to reach the leftmost point of its motion? Angular Frequency Angular frequency is another way to measure the frequency of an oscillation. Angular frequency is angular displacement of the sinusoidal function divided by the period. This is synonymous with angular velocity. The benefit of using angular frequency as opposed to frequency is that it keeps 2π from popping up over and over again. The units for angular frequency are radians/sec. ω = 2π T = 2πf 3
Simple Harmonic Motion A graph or a function that has the form of a sine or cosine function is called sinusoidal. A sinusoidal oscillation is called simple harmonic motion (SHM). SHM is caused by a linear restoring force. This means that the net force is always pointing towards equilibrium and is proportional to the displacement from equilibrium. Equation of SHM SHM is modeled by a sine or cosine function. The standard form of this type of function is shown below. x t = Acos ωt + φ 4
Examples of SHM The two examples of simple harmonic motion that we will study are masses on springs and pendulums. We will analyze the period and frequency of both situations as well as the energy in each kind of oscillation. Mass-Spring Systems 5
Springs and Hooke s Law If we displace a glider (meaning no friction) attached to a spring from its equilibrium position, the spring exerts a restoring force back toward equilibrium. This is a linear restoring force; the net force is toward the equilibrium position and is proportional to the distance from equilibrium. For springs, this force is called Hooke s Law. x is the displacement from equilibrium, and k is called the spring constant. Spring constant represents the stiffness of the spring and has units of N/m. F = k x Example A scale used to weigh fish consists of a spring hung from a support. The spring s equilibrium length is 10.0 cm. When a 4.0 kg fish is suspended from the end of the spring, it stretches to a length of 12.4 cm. a. What is the spring constant k for this spring? b. If an 8.0 kg fish is suspended from the spring, what will be the length of the spring? 6
Amplitude of a Mass-Spring Oscillator The amplitude A is the object s maximum displacement from equilibrium. Oscillation about an equilibrium position with a linear restoring force is always simple harmonic motion. Frequency and Period of a Mass-Spring Oscillator The frequency and period of an oscillating massspring system depends on the mass of the block and the spring constant of the spring. Notice that the period and frequency do not depend of the amplitude of the oscillator. 7
Example A mass oscillates on a horizontal spring with period T 2.0 s. If the amplitude of the oscillation is doubled, what will the new period will be? If a second identical block is glued to the top of the first block, what will the new period will be? Example A spring has an unstretched length of 10.0 cm. A 25 g mass is hung from the spring, stretching it to a length of 15.0 cm. If the mass is pulled down and released so that it oscillates, what will be the frequency of the oscillation? 8
Modeling Position for SHM Consider the motion of a mass-spring system. We have spent time discussing the forces and energy related to such a system, but now we will shift our focus towards modeling the position as a function of time. We will begin by analyzing the forces on the block using Newton s second law. F = ma kx = ma kx = m d x dt k x m = d x dt Mass-Spring Systems This is generally as far as the AP test will want you to go in a problem (they will say write but do not solve a differential equation that models this motion). We now have what is called a second order linear differential equation. There are different ways to find solutions to such an equation, but the best one here is what is called the guess and check method. I know, sounds legit. x k m = d x dt What function can I take two derivatives of and get something close to the original function? A cosine perhaps? Lets check and see if we are right. 9
Mass-Spring Systems x t = Acos ωt + φ dx = v t dt = Aω sin ωt + φ d x dt = a t = Aω cos ωt + φ d x dt = ω Acos ωt + φ d x dt = ω x We were right! In addition, to determining the position function, we also have functions for the velocity and acceleration. This also shows that the angular frequency of this function is determined by physical characteristics of the block and spring (mass and spring constant). ω = k m ω = k m Period of a Spring We can use this angular frequency and its relation period to determine the period for a mass spring system. Interestingly enough, we find that both the angular frequency and period of the oscillator is the same as uniform circular motion with the same radius. T = 2π ω T = 2π m k 10
Pendulums Simple Pendulums A simple pendulum is a mass suspended from a pivot point by a light string or rod. The mass moves along a circular arc. The net force is the tangential component of the weight. The force on a pendulum is a linear restoring force for small angles, so the pendulum will undergo simple harmonic motion. 11
Frequency and Period of a Pendulum The period and frequency of a pendulum depends on the length of the string and the acceleration due to gravity. Notice that the period and frequency do not depend on the mass of the pendulum or the amplitude (assuming that the amplitude is small). Example A ball on a massless, rigid rod oscillates as a simple pendulum with a period of 2.0 s. If the ball is replaced with another ball having twice the mass, what will the period will be? If the pendulum is taken to the moon of Planet X, where the freefall acceleration g is half as big, what will the period will be? 12
Differential Equations We will use Newton s second law for rotation to derive a angular position function. τ = Iα mglsin θ gsin θ g sin θ l = = ml d θ dt = l d θ dt d θ dt There is a problem here. The differential equation is second order, but the sine term makes it nonlinear. Solutions to non-linear differential equations require initial values and direct computation. We will confine this scenario to small angles, < 20 degrees, so that we can use the small angle approximation. If θ 1, then sin θ θ Differential Equations This linearizes our differential equation, making it solvable using our guess and check method. g l θ = d θ dt This gives us the same solution that we had with an oscillating mass-spring system θ t = Acos ωt + φ ω = g l We can also use our relationship between angular frequency and period to find the period of a pendulum for small oscillations. T = 2π ω ω = g l T = 2π l g 13
Physical Pendulums A physical pendulum is an oscillating body that rotates according to the location of its center of mass rather than a simple pendulum where all the mass is located at the end of a light string. A rod or a meter stick would be examples of physical pendulums. The difference mathematically is that the gravitational torque is smaller, and the moment of inertia will be different. Generally speaking, we will define the period of a physical pendulum as: T = 2π I mgd I = moment of inertia d = distance from pivot to center of mass (moment arm) Torsion Pendulums Torsion pendulums are oscillating discs suspended from a wire. The torsion pendulum has simple harmonic behavior like a simple pendulum. θ t = Acos ωt + φ 14
Energy in SHM Energy in SHM For an oscillator in SHM, the total mechanical energy is constant (assuming no friction or air resistance), and the potential and kinetic energy continuously change. For a mass-spring system, the energy graph is represented by a parabola. We often refer to this as an energy well. The potential energy stored in a spring is: U = 1 kx 2 15
Example A 0.2 kg block is connected to a horizontal spring on a frictionless surface. The block is stretched a distance of 20 cm and the spring constant is k = 300 N/m. a. How much elastic potential energy is stored in the spring? b. How much is work is done on the block-spring system in stretching the spring? c. What is the maximum speed of the block? d. Calculate the period and frequency of the block. Non-linear Springs Non-linear springs do not obey Hooke s law. This is due to the irregular geometry of the spring. The spring potential energy must be derived using work-energy theorem. Non-linear springs do not oscillate with SHM. 16
Example A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement: F = kx. This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released. Determine each of the following in terms of k, A, and M: a. The potential energy in the spring at the instant the mass is released b. The maximum speed of the mass c. The displacement of the mass at the point where the potential energy of the spring and the kinetic energy of the mass are equal The amplitude of the oscillation is now increased: d. State whether the period of the oscillation increases, decreases, or remains the same. Justify your answer. 17