1. Physics 1B03summer - Lecture 7 Test 2 Tuesday June 2 at 9:30 am CNH-104 Energy, Work, Power, Simple Harmonic Motion Class will start at 11am

2. Physics 1B03summer - Lecture 7 Energy of a SHO

3. Physics 1B03summer - Lecture 7 Energy of a SHO Recall that for a spring: E Tot = K + U = 1/2mv 2 + 1/2kx 2 And we know that: x=A cos(ωt+φ) v=-A ω sin(ωt+φ)

4. Physics 1B03summer - Lecture 7 Energy in SHM M Look again at the block & spring We could also write E = K+U = ½ m(v max ) 2 Hence: E Tot = ½kA 2

5. Physics 1B03summer - Lecture 7 Suppose you double the amplitude of the motion, what happens to the total energy? a)Doubles b)4 x Larger c)Doesn’t change Quiz

6. Physics 1B03summer - Lecture 7 a)Doubles b)4 x Larger c)Doesn’t change Quiz Suppose you double the amplitude of the motion, what happens to the maximum speed ?

7. Physics 1B03summer - Lecture 7 Suppose you double the mass and amplitude of the object, what happens to the maximum acceleration ? a)Doubles b)4 x Larger c)Doesn’t change Quiz

8. Physics 1B03summer - Lecture 7 Energy Since we know the total energy of a SHM, we can calculate the or displacement velocity at any point in time: E Tot = ½ kA 2 = K+U = ½ mv 2 + ½ kx 2 So, if x=0, all E is in kinetic, and v is at max if x=A, all E is in potential, and v is zero

9. Physics 1B03summer - Lecture 7 A 100g block is 5cm from the equilibrium position moving at 1.5m/s. The angular frequency is 2 rad/s. a) What is the total energy of the system ? b) What is the amplitude of the oscillations ? Example 1

10. Physics 1B03summer - Lecture 7 Example 2 A 500g block on a spring is pulled 20cm and released. The motion has a period of 0.8s. What is the velocity when the block is 15.4cm from the equilibrium ?

11. Physics 1B03summer - Lecture 7 A 1.0kg block is attached to a spring with k=16N/m. While the block is at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40cm/s. a)what is the amplitude of the subsequent oscillations ? b)what is the block’s speed at the point where x=A/2 ? Example 3

12. Physics 1B03summer - Lecture 7 10 min rest

13. Physics 1B03summer - Lecture 7 SHM and Circular Motion  A Uniform circular motion about in the xy -plane, radius A, angular velocity  :  (t) =  0 +  t (similar to, x=x o +vt) and so Hence, a particle moving in one dimension can be expressed as an ‘imaginary’ particle moving in 2D (circle), or vice versa - the ‘projection’ of circular motion can be viewed as 1D motion.

14. Physics 1B03summer - Lecture 7 Compare with our expression for 1-D SHM. Result: SHM is the 1-D projection of uniform circular motion.

15. Physics 1B03summer - Lecture 7 Phase Constant, θ o  A  For circular motion, the phase constant is just the angle at which the motion started.

16. Physics 1B03summer - Lecture 7 Example An object is moving in circular motion with an angular frequency of 3π rad/s, and starts with an initial angle of π/6. If the amplitude is 2.0m, what is the objects angular position at t=3sec ? What are the x and y values of the position at this time ?

17. Physics 1B03summer - Lecture 7 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

18. Physics 1B03summer - Lecture 7 Simple pendulum: SHM: The pendulum is not a simple harmonic oscillator! However, take small oscillations: (radians) if  is small. Then Compare:

19. Physics 1B03summer - Lecture 7 Simple Pendulum Using sin(θ)~θ for small angles, we have the following equation of motion: With: Hence: Application - measuring height - finding variations in g → underground resources or: -------------------------------------------------------------------------

20. Physics 1B03summer - Lecture 7 This looks like For small  :, with angle  instead of x. So, the position is given by amplitude phase constant (2  / period)

21. Physics 1B03summer - Lecture 7 a) longer. b) same. c) shorter. Quiz: 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 (by how much, in percent?)

22. Physics 1B03summer - Lecture 7 Application 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? ?

23. Physics 1B03summer - Lecture 7 Quiz 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?

24. Physics 1B03summer - Lecture 7 A physicist wants to know the height of a building. He notes that a long pendulum that extends from the ceiling almost to the floor has a period of 12 s. How tall is the building? Example

25. Physics 1B03summer - Lecture 7 Compare Springs and Pendulum Newton’s 2 nd Law:

26. Physics 1B03summer - Lecture 7 t x x t SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time. SHM and Damping – EXTRA !!!

27. Physics 1B03summer - Lecture 7 For weak damping (small b), the solution is: f =  bv where b is a constant damping coefficient x t A damped oscillator has external nonconservative force(s) acting on the system. A common example is a force that is proportional to the velocity. eg: green water A e -(b/2m)t F=ma give:

28. Physics 1B03summer - Lecture 7 10 min rest

29. Physics 1B03summer - Lecture 7 Wave Motion – Chapter 20 Qualitative properties of wave motion Mathematical description of waves in 1-D Sinusoidal waves

30. Physics 1B03summer - Lecture 7 A wave is a moving pattern. For example, a wave on a stretched string: The wave speed v is the speed of the pattern. No particles move at this wave speed – it is the speed of the wave. However, the wave does carry energy and momentum. v Δx = v Δt time t t +Δt

31. Physics 1B03summer - Lecture 7 The wave moves this way The particles move up and down If the particle motion is perpendicular to the direction the wave travels, the wave is called a “transverse wave”. Transverse waves Examples: Waves on a string; waves on water; light & other electromagnetic waves; some sound waves in solids (shear waves)

32. Physics 1B03summer - Lecture 7 The wave moves long distances parallel to the particle motions. Longitudinal waves Example: sound waves in fluids (air, water) Even in longitudinal waves, the particle velocities are quite different from the wave velocity. The speed of the wave can be orders of magnitude larger than the particle speeds. The particles move back and forth.

33. Physics 1B03summer - Lecture 7 Quiz wave motion A B C Which particle is moving at the highest speed? A) A B) B C) C D) all move with the same speed

34. Physics 1B03summer - Lecture 7 For these waves, the wave speed is determined entirely by the medium, and is the same for all sizes & shapes of waves. eg. stretched string: (A familar example of a dispersive wave is an ordinary water wave in deep water. We will discuss only non-dispersive waves.) Non-dispersive waves: the wave always keeps the same shape as it moves.

35. Physics 1B03summer - Lecture 7 Principle of Superposition When two waves meet, the displacements add: So, waves can pass through each other: v v (for waves in a “linear medium”)

36. Physics 1B03summer - Lecture 7 Quiz: “equal and opposite” waves v v Sketch the particle velocities at the instant the string is completely straight. v v

37. Physics 1B03summer - Lecture 7 Reflections Waves (partially) reflect from any boundary in the medium: 1) “Soft” boundary: Reflection is upright light string, or free end

38. Physics 1B03summer - Lecture 7 Reflection is inverted Reflections 2) “Hard” boundary: heavy rope, or fixed end.

39. Physics 1B03summer - Lecture 7 Quiz A) y=  f(x  vt) B) y=  f(x+vt) C) y=  f(  x  vt) D) y=  f(  x+vt) A wave show can be described by the equation y=f(x  vt), as it travels along the x axis. It reflects from a fixed end at the origin. The reflected wave is described by:

40. Physics 1B03summer - Lecture 7 The math: Suppose the shape of the wave at t = 0 is given by some function y = f(x). y y = f (x) v at time t : y = f (x - vt) y vt at time t = 0 : Note: y = y(x,t), a function of two variables; f is a function of one variable

41. Physics 1B03summer - Lecture 7 Non-dispersive waves: y (x,t) = f (x ± vt) + sign: wave travels towards –x  sign: wave travels towards +x f is any (smooth) function of one variable. eg. f(x) = A sin (kx)