Physics 121 April 8 2008 Harmonic Motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 April 8 2008 Course Information Topics to be discussed today Simple Harmonic Motion Review Simple Harmonic Motion Example Systems Damped Harmonic Motion Driven Harmonic Motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 April 8 2008 Homework set 8 is due on Saturday morning April 12 at 8 30 am Homework set 9 will be available on Saturday morning at 8 30 am and will be due on Saturday morning April 19 at 8 30 am Requests for regarding part of Exam 1 and 2 need to be given to me by April 17 You need to write down what I should look at and give me your written request and your blue exam booklet s Frank L H Wolfs Department of Physics and Astronomy University of Rochester 1 Harmonic motion a quick review Motion that repeats itself at regular intervals Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion a quick review Phase Constant Amplitude x t x m cos t Angular Frequency Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion a quick review Other variables frequently used to describe simple harmonic motion The period T the time required to complete one oscillation The period T is equal to 2 The frequency of the oscillation is the number of oscillations carried out per second 1 T The unit of frequency is the Hertz Hz Per definition 1 Hz 1 s 1 Frank L H Wolfs Department of Physics and Astronomy University of Rochester 2 Simple Harmonic Motion a quick review What forces are required Using Newton s second law we can determine the force responsible for the harmonic motion F ma m 2x The total mechanical energy of a system carrying out simple harmonic motion is constant A good example of a force that produces simple harmonic motion is the spring force F kx The angular frequency depends on both the spring constant k and the mass m k m Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The torsion pendulum What is the angular frequency of the SHM of a torsion pendulum When the base is rotated it twists the wire and a the wire generated a torque which is proportional to the the angular twist K The torque generates an angular acceleration d2 dt 2 I K I The resulting motion is harmonic motion with an angular frequency K I Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The simple pendulum Calculate the angular frequency of the SHM of a simple pendulum A simple pendulum is a pendulum for which all the mass is located at a single point at the end of a massless string There are two forces acting on the mass the tension T and the gravitational force mg The tension T cancels the radial component of the gravitational force Frank L H Wolfs Department of Physics and Astronomy University of Rochester 3 Simple Harmonic Motion SHM The simple pendulum The net force acting on he mass is directed perpendicular to the string and is equal to F mg sin The minus sign indicates that the force is directed opposite to the angular displacement When the angle is small we can approximate sin by F mg mg x L Note the force is again proportional to the displacement Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The Simple Pendulum The equation of motion for the pendulum is thus F m d 2x dt 2 mg L x or d2x dt2 g L x The equation of motion is the same as the equation of motion for a SHM and the pendulum will thus carry out SHM with an angular frequency g L The period of the pendulum is thus 2 2 L g Note the period is independent of the mass of the pendulum Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The physical pendulum In a realistic pendulum not all mass is located at a single point The motion carried out by this realistic pendulum around its rotation point O can be determined by determining the total torque with respect to this point mgh sin If the angle is small we can approximate the torque by mgh Frank L H Wolfs Department of Physics and Astronomy University of Rochester 4 Simple Harmonic Motion SHM The physical pendulum The angular acceleration is related to the torque I The equation of motion for the angular acceleration is given by d 2 mgh 2 dt I I This again is an equation for SHM with an angular frequency where 2 mgh I Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The physical pendulum The period of the pendulum is equal to T physical 2 I 2 mgh We can double check our answer by requiring that the simple pendulum is a special case of the physical pendulum h L I mL2 T 2 Frank L H Wolfs I mL2 L 2 2 mgh mgL g Department of Physics and Astronomy University of Rochester Physics 121 Quiz lecture 21 The quiz today will have 3 questions Frank L H Wolfs Department of Physics and Astronomy University of Rochester 5 Simple Harmonic Motion SHM The equation of motion All examples of SHM were derived from he following equation of motion d2x 2 x dt 2 The most general solution to the equation is x t A cos t B sin t Frank L H Wolfs Department of Physics and Astronomy University of Rochester Simple Harmonic Motion SHM The equation of motion If A B x t A cos t B sin t 1 A sin t sin t 2 1 1 2A sin cos t 4 4 2 2 2 2 which is SHM Frank L H Wolfs Department of Physics and Astronomy University of Rochester Damped Harmonic Motion Consider what happens when in addition to the restoring force a damping force such as the drag force is acting on the system F kx b dx dt The equation of motion is now given by d 2 x b dx k x 0 dt 2 m dt m Frank L H Wolfs Department of Physics and Astronomy University of Rochester 6 Damped Harmonic Motion The general solution of this equation of motion is x t Aei t If we substitute this solution in the equation of motion we find 2 Aei t i b i t k Ae Aei t 0 m m In order to satisfy the equation of motion the angular frequency must satisfy the following condition 2 i b k 0 m m Frank L H Wolfs Department of Physics and Astronomy University of Rochester Damped Harmonic Motion We can solve this equation and determine the two possible values of the angular velocity 1 i …