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

Concept Question: SHO and the Pendulum Solution 2: The angular frequency is a constant of the motion and by definition is ω 0 = 2π/T. For small angle the pendulum approximates a simple harmonic oscillator with ω 0 = (g/l) 1/2. The angular speed ω by definition is the magnitude of the component of the angular velocity, ω z = dθ/dt. Note that sometimes the symbol ω may be used for both quantities. This is a result of the fact that for uniform circular motion, angular frequency and angular speed are equal because the period T = 2πR/v and the speed and angular speed are related by v = Rω. Therefore T = 2π/ω. So for this special case ω = ω 0. 9

Demonstration Pendulum: Amplitude Effect on Period 10

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 . 11

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. 12

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? 13

Concept Question: Physical Pendulum Answer 3. When the disk is fixed to the rod, an internal torque will cause the disk to rotate about its center of mass. When the pendulum reaches the bottom of its swing, the decrease in potential energy will be result in an increase in the rotational kinetic energy of both the rod and the disk and the center of mass translation kinetic energy of the rod-disk system. When the disk is mounted on the frictionless bearing there is no internal torque that will make the disk start to rotate about its center of mass when the pendulum is released. Therefore when the pendulum reaches the bottom of its swing, the same decrease in potential energy will be transferred into a larger smaller in rotational kinetic energy of just the rod since the disc is not rotating and a greater increase in the center of mass translation kinetic energy of the rod-disk system. So when the disk bearings are frictionless, the center of mass of the rod-disk system is traveling faster at the bottom of its arc hence will take less time to complete one cycle and so the period is shorter compared to the fixed disk. 14

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. 16

Small Oscillations 17

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 18

Concept Question: Energy Diagram 1 Solution 1. Because the energy is greater than the potential energy for values of x that approach negative infinity (on the left in the above figure), the particle can escape to infinity in the negative x-direction with a positive kinetic energy. 19

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 20

Concept Question: Energy Diagram 2 Solution 5. Now the range of motion for the particle is limited to the regions in which the kinetic energy is either zero or positive, so the particle is confined to the regions where the potential energy is less than the energy. Hence the particle periodically revisits a and b. 21

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 22

Concept Question: Energy Diagram 3 Solution 6. Now the range of motion for the particle is limited to the regions in which the kinetic energy is either zero or positive, so the particle is confined to either the region around a and or the region around b but since we do not know where the particle has started, we do not have enough information to state where the particle will be. 23

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 24

Concept Question: Energy Diagram 4 Solution 3. Now the range of motion for the particle is limited to the regions in which the kinetic energy is either zero or positive, so the particle is confined to move around the region surrounding a. The motion will be periodic but not simple harmonic motion because the potential energy function is not a quadratic function and only for quadratic potential energy functions will the motion be simple harmonic. Hence the particle oscillates around a. 25

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 26

Concept Question: Energy Diagram 5 Solution 2. Now the particle oscillates around the region surrounding a. Since the energy is so close to the minimum of the potential energy, we can approximate the potential energy as a quadratic function and hence the particle motion approximates simple harmonic motion. 27

Small Oscillations 28 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

Table Problem: Small Oscillations 29 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 30

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