an angular simple harmonic oscillator

1. An AngularSimple Harmonic Oscillator The figure shows an angular version of a simple harmonic oscillator In this case the mass rotates around it�s center point and twists the suspending wire This is called a torsional pendulum with torsion referring to the twisting motion

3. Pendulums When we were discussing the energy in a simple harmonic system, we talked about the �springiness� of the system as storing the potential energy But when we talk about a regular pendulum there is nothing �springy� � so where is the potential energy stored?

4. The Simple Pendulum As we have already seen, the potential energy in a simple pendulum is stored in raising the bob up against the gravitational force The pendulum bob is clearly oscillating as it moves back and forth � but is it exhibiting SHM?

5. Going back to our definition of torque, we can see that the restoring force is producing a torque around the pivot point of:

6. The Simple Pendulum This doesn�t appear too promising until we make the following assumption � that ? is small� If ? is small we can use the approximation that sin ? ? ? (as long as we remember to express ? in radians)

7. Making the substitution we then get:

8. The Simple Pendulum where I is the moment of inertia of the pendulum

9. The Physical Pendulum Now suppose that the mass is not all concentrated in the bob? In this case the equations are exactly the same, but the restoring force acts through the center of mass of the body (C in the diagram) which is a distance h from the pivot point

10. The Physical Pendulum So we go back to our previous equation for the period and replace L with h to get:

11. The Physical Pendulum The other difference in this case is that the rotational inertia will not be a simple I = mL2 but rather something more complicated which will depend on the shape of the body

12. The Physical Pendulum For any physical pendulum that oscillates around a point O with period T, there is a simple pendulum of length L0 which oscillates with the same period The point P on the physical pendulum a distance L0 from O is called the center of oscillation

13. Measuring g The equation for the period of a physical pendulum gives us a very nice and neat relationship between T, I and g

14. Suppose we have a uniform rod of length L which we allow to rotate from one end as shown the moment of inertia I =mL2/3 h = L/2

15. Sample Problem 1 A 1 meter stick swings about a pivot point at one end at a distance h from its center of mass What is the period of oscillation?

16. Sample Problem 1

17. Sample Problem 2 What is the distance L0 between the pivot point of the stick and the center of oscillation of the stick?

27. At the right we have a plot of data recorded by Galileo of an object (the moon Callisto) that moved back and forth relative to the disk of Jupiter

28. The circles are Galileo�s data points and the curve is a best fit to that data This would strongly suggest that Callisto exhibits SHM

29. But in fact Callisto is moving with pretty much a constant speed in a nearly circular orbit about Jupiter So what is it that we are seeing in the data?

30. SHM is the projection of uniform circular motion on a diameter of the circle in which the latter motion takes place

31. Simple Harmonic Motion & Uniform Circular Motion We have at the right a reference circle; the particle at point P� is moving on that circle at a constant angular speed ? The radius of our reference circle is xm Finally, the projection of the position of P� onto the x axis is the point P

32. Simple Harmonic Motion & Uniform Circular Motion We can easily see that the position of the projection point P is given by the formula:

33. Simple Harmonic Motion & Uniform Circular Motion Similarly, if we look at the velocity of our particle (and use the relationshipv = ?r) we can see that it obeys:

34. Simple Harmonic Motion & Uniform Circular Motion And finally, if we look at the acceleration of our particle (and use the relationship ar = ?2r) we can see that it obeys: