Physics 1D03 - Lecture 341 Harmonic Motion ( III ) Simple and Physical Pendulum SHM and uniform circular motion

Physics 1D03 - Lecture 342 Simple Pendulum Gravity is the “restoring force” taking the place of the “spring” in our block/spring system. Instead of x, measure the displacement as the arc length s along the circular path. Write down the tangential component of F=ma: L θ T mg sin θ Restoring force s mg

Physics 1D03 - Lecture 343 Simple pendulum: SHM: The pendulum is not a simple harmonic oscillator! However, take small oscillations: (radians) if  is small. Then

Physics 1D03 - Lecture 344 This looks like For small  :, with angle  instead of x. The pendulum oscillates in SHM with an angular frequency and the position is given by amplitude phase constant (2  / period)

Physics 1D03 - Lecture 345 a simple harmonic oscillator is a mathematical ‘approximation’ to the full problem for large amplitudes, the solution that the SHO gives us start to deviate from what they actually should be : Unlike the SHO, the actual solution depends on the amplitude!

Physics 1D03 - Lecture 346 Quiz Pendulum clocks (“grandfather clocks”) often have a swinging arm with an adjustable weight. Suppose the arm oscillates with T=1.05sec and you want to adjust it to 1.00sec. Which way do you move the weight? ? A) Up B) Down

Physics 1D03 - Lecture 347 Question: A simple pendulum hangs from the ceiling of an elevator. If the elevator accelerates upwards, the period of the pendulum: a)Gets shorter b)Gets larger c)Stays the same Question: What happens to the period of a simple pendulum if the mass m is doubled?

Physics 1D03 - Lecture 348 a)longer b)shorter Question: How high is the ceiling? Question: A geologist is camped on top of a large deposit of nickel ore, in a location where the gravitational field is 0.01% stronger than normal. the period of his pendulum will be (and by how much, in percent?)

Physics 1D03 - Lecture 349 Simple pendulum : A particle on a massless string.   P CM mg “ Physical” pendulum : any rigid body, pivoted at P, and free to swing back and forth. To find the period: 1) Consider the torque due to gravity 2) Write  (  )  I  I (d 2  / dt 2 ) 3) SHM if  is proportional to 

Physics 1D03 - Lecture 3410 Calculate torque about the end: Example: a metre stick, pivoted at one end. What is its period of oscillation? “Uniform thin rod, pivot at end”: I = 1 / 3 ML 2 and so Note, this does not describe SHM!  Mg L

Physics 1D03 - Lecture 3411 But, for small oscillations, sin  This is like a simple pendulum of length 2 / 3 L. so The angular frequency is and the period is This looks like, with angle  instead of x.

Physics 1D03 - Lecture 3412 For a simple pendulum, I = Ml 2 and d θ Mg in general Any swinging object can be analysed in a similar way; we just need to know its moment of inertia, I, about the pivot point. amplitude phase constant (2  / period)

Physics 1D03 - Lecture 3413 Quiz: What happens to the period of the metre stick when the pivot is moved closer to the centre? A)The period gets longer. B)The period gets shorter. C)The period stays the same. D)It’s rather difficult to tell. mg P

Physics 1D03 - Lecture 3414, or I is the moment of inertia about the pivot. From the parallel-axis theorem: (for a uniform thin rod). So (for a uniform thin rod).

Physics 1D03 - Lecture 3415 SHM and Circular Motion  A Uniform circular motion about in the xy plane, radius A, speed v, and angular velocity  = v/A :  (t) =  0 +  t and so Real particle moves on the x axis “Imaginary” particle moves in a circle.

Physics 1D03 - Lecture 3416 Compare with our expression for 1-D SHM. Result: An interesting side problem: (try this out on your own) Start with Differentiate twice, and then show SHM is the 1-D projection of uniform circular motion.

Physics 1D03 - Lecture 3417 Summary The projection of uniform circular motion onto an axis is SHM in 1-D. The oscillation of a simple pendulum is approximately SHM, if the amplitude is small, with angular frequency

Physics 1D03 - Lecture 3418 Practice problems, Chapter 15 3, 5, 11, 19, 23, 31, 67 (6 th ed – Chapter 13) 1, 3, 5, 9, 19, 23, 29, 67