Torque and Simple Harmonic Motion Week 13D2 Today’s Reading Assignment Young and Freedman:

Announcements Problem Set 11 Due Thursday Nov 1 9 pm Sunday Tutoring in from 1-5 pm W013D3 Reading Assignment Young and Freedman:

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 Angle of oscillation is small Simple harmonic oscillator Analogy to spring equation Angular frequency of oscillation Period

Simple Pendulum: Approximation to Exact Period Equation of motion: Approximation to exact period: Taylor Series approximation: 7

Concept Question: SHO and the Pendulum Suppose the point-like object of a simple pendulum is pulled out at by an angle   << 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 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  At t = 0 the disk is released from rest at an angular displacement of . Find the subsequent time dependence of the angular displacement . 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 d cm 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 Small angle approximation Equation of motion Angular frequency Period

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 16 Potential energy function for object of mass m Motion is limited to the region Potential energy has a minimum at Small displacement from minimum, approximate potential energy by Angular frequency of small oscillation

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 22 A particle of effective mass m is acted on by a potential energy given by where and are positive constants a)Sketch as a function of. b)Find the points where the force on the particle is zero. Classify them as stable or unstable. Calculate the value of at these equilibrium points. c)If the particle is given a small displacement from an equilibrium point, find the angular frequency of small oscillation.

Appendix 23

Simple Pendulum: Mechanical Energy Velocity Kinetic energy Initial energy Final energy Conservation of energy

Simple Pendulum: Angular Velocity Equation of Motion Angular velocity Integral form Can we integrate this to get the period?

Simple Pendulum: Integral Form Change of variables “Elliptic Integral” Power series approximation Solution

Simple Pendulum: First Order Correction Period Approximation First order correction