#### Transcript Simple Pendulum

Torque and Simple Harmonic Motion 8.01 Week 13D2 Today’s Reading Assignment Young and Freedman: 14.1-14.6 1 Announcements Problem Set 11 Due Thursday Nov 1 9 pm Sunday Tutoring in 26-152 from 1-5 pm W013D3 Reading Assignment Young and Freedman: 14.1-14.6 2 Simple Pendulum Table Problem: Simple Pendulum by the Torque Method (a) Find the equation of motion for θ(t) using the torque method. (b) Find the equation of motion if θ is always <<1. Table Problem: Simple Pendulum by the Energy Method 1. Find an expression for the mechanical energy when the pendulum is in motion in terms of θ(t) and its derivatives, m, l, and g as needed. 2. Find an equation of motion for θ(t) using the energy method. Simple Pendulum: Small Angle Approximation Equation of motion 2 d lmg sin ml 2 2 dt Angle of oscillation is small sin Simple harmonic oscillator d 2 g 2 l dt d 2x k x 2 m dt Analogy to spring equation Angular frequency of oscillation Period T0 2 0 0 g / l 2 l / g Simple Pendulum: Approximation to Exact Period Equation of motion: 2 d lmg sin ml 2 2 dt Approximation to exact period: T ; T0 (sin0 / 0 )3/8 T0 T Taylor Series approximation: T ; T0 1 2 0 16 7 Concept Question: SHO and the Pendulum Suppose the point-like object of a simple pendulum is pulled out at by an angle 0 << 1 rad. Is the angular speed of the point-like object equal to the angular frequency of the pendulum? 1.Yes. 2.No. 3.Only at bottom of the swing. 4.Not sure. 8 Demonstration Pendulum: Amplitude Effect on Period 9 Table Problem: Torsional Oscillator A disk with moment of inertia I cm about the center of mass rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium position by an angle , the rod exerts a restoring torque given by cm At t = 0 the disk is released from rest at an angular displacement of 0 . Find the subsequent time dependence of the angular displacement (t) . 10 Worked Example: Physical Pendulum A general physical pendulum consists of a body of mass m pivoted about a point S. The center of mass is a distance dcm from the pivot point. What is the period of the pendulum. 11 Concept Question: Physical Pendulum A physical pendulum consists of a uniform rod of length l and mass m pivoted at one end. A disk of mass m1 and radius a is fixed to the other end. Suppose the disk is now mounted to the rod by a frictionless bearing so that is perfectly free to spin. Does the period of the pendulum 1. increase? 2. stay the same? 3. decrease? 12 Physical Pendulum Rotational dynamical equation r r S IS Small angle approximation sin Equation of motion Angular frequency Period lcm mg d 2 2 IS dt 0 2 lcm mg IS IS T 2 0 lcm mg Demo: Identical Pendulums, Different Periods Single pivot: body rotates about center of mass. Double pivot: no rotation about center of mass. 14 Small Oscillations 15 Small Oscillations Potential energy function U (x) for object of mass m Motion is limited to the region x1 x x2 Potential energy has a minimum at x x0 Small displacement from minimum, approximate potential energy by 2 1 d U U (x) ; U (x0 ) (x x0 )2 2 (x0 ) 2! dx 1 U (x) ; U (x0 ) keff (x x0 )2 2 Angular frequency of small oscillation 0 keff / m d 2U (x0 ) / m 2 dx 16 Concept Question: Energy Diagram 1 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity in the – x-direction 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information 17 Concept Question: Energy Diagram 2 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information 18 Concept Question: Energy Diagram 3 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information 19 Concept Question: Energy Diagram 4 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information 20 Concept Question: Energy Diagram 5 A particle with total mechanical energy E has position x > 0 at t = 0 1) escapes to infinity 2) approximates simple harmonic motion 3) oscillates around a 4) oscillates around b 5) periodically revisits a and b 6) not enough information 21 Table Problem: Small Oscillations A particle of effective mass m is acted on by a potential energy given by x 2 x 4 U (x) U 0 2 x0 x0 where U 0 and x0 are positive constants a)Sketch U (x) / U 0 as a function of x / x. 0 b)Find the points where the force on the particle is zero. Classify them as U (x) / Uat0 these equilibrium stable or unstable. Calculate the value of points. c)If the particle is given a small displacement from an equilibrium point, find the angular frequency of small oscillation. 22 Appendix 23 Simple Pendulum: Mechanical Energy Velocity vtan d l dt 2 Kinetic energy 1 2 1 d K f mvtan m l 2 2 dt Initial energy E0 K 0 U 0 mgl(1 cos0 ) 2 Final energy 1 d E f K f U f m l mgl(1 cos ) 2 dt Conservation of energy 2 1 d m l mgl(1 cos ) mgl(1 cos0 ) 2 dt Simple Pendulum: Angular Velocity Equation of Motion Angular velocity d 2g (cos cos 0 ) dt l Integral form d (cos cos 0 ) 2g dt l Can we integrate this to get the period? Simple Pendulum: Integral Form Change of variables “Elliptic Integral” Power series approximation 1 b sin a 2 Solution 2 1 2 b sin 2 bsin a sin 2 1 cos sin 2 2 2 0 da 1 b sin a 2 2 12 1 2 2 3 4 4 1 b sin a b sin a 2 8 1 2 sin 2 0 2 2 g T l g dt l Simple Pendulum: First Order Correction Period l T 2 g 2 1 2 1 2 1 4 sin 0 2 1 4 sin 0 2 Approximation First order correction sin 2 0 2 02 4 l T 2 g 1 2 1 2 1 16 0 T0 1 16 0 1 2 T0 T1 where T1 0 T0 16