Chap. 15: Simple Harmonic Motion

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1 Chap. 15: Simple Harmonic Motion Announcements: CAPA is due next Tuesday and next Friday. Web page:

2 Examples of periodic motion vibrating guitar string A pendulum swinging back and forth quartz oscillator in a watch molecular vibrations

3 Periodic motion In periodic motion systems there is an equilibrium point at which the net force is 0 so if the system is at rest at that point it will not move (it is in equilibrium). Any movement away from the equilibrium point results in a force toward the equilibrium point. This is called a restoring force. We will investigate two examples: A mass at the end of a spring (restoring force from spring) A mass at the end of a string pendulum (restoring force is gravity) Both situations are examples of simple harmonic motion which occur when the restoring force is proportional to the displacement. It is again helpful for many calculations to use radians rather than degrees

4 Clicker question 1 A particle which can move along the x-axis is initially at equilibrium at the origin. When displaced from the origin, the particle experiences a restoring force proportional to the displacement from the origin. Which of these equations represents this situation? (in these equations, k is a positive constant) A. B. C. D. E. O x

5 Clicker question 1 A particle which can move along the x-axis is initially at equilibrium at the origin. When displaced from the origin, the particle experiences a restoring force proportional to the displacement from the origin. Which of these equations represents this situation? (in these equations, k is a positive constant) A. B. C. D. E. x O Since the restoring force is proportional to the displacement it must be of the form k*x where k is the proportionality constant. A restoring force acts in the direction of the equilibrium point (which is the origin in this case) and k is positive so

6 Mass on end of spring We previously encountered springs as a source for elastic potential energy and we found the following: 0 The force exerted by the spring is. The potential energy stored in a spring is. The maximum force (and therefore maximum acceleration) occurs at the endpoints and there is no force (but maximum velocity) when the particle goes through the equilibrium point. We did not investigate how the position varies as a function of time. Can we use Newton s equations derived for constant acceleration? Since the acceleration is not constant, cannot use constant acceleration equations

7 Newton s 2 nd law Mass on end of spring can be written The mass on the end of a spring experiences a force Setting these equations equal to each other: Dividing both sides by m: This is a differential equation for x. Finding the solution to this means we find x, which will vary with t. The secret to solving differential equations is guessing the answer and proving it works. We are going to guess that

8 Clicker question 2 What is for? A. B. C. D. E.

9 Clicker question 2 What is for? A. B. C. D. First, need to know Second, need chain rule: E. Can also calculate acceleration

10 Mass on a spring We want a solution to the equation We now know that for So this is a solution as long as Note that the constants A and are not determined by the equations of motion. They will be determined by the initial conditions of the problem.

11 Clicker question 3 I have a heavy mass and a light mass, each hanging on a similar spring. What can you say about the oscillation frequencies? A. Heavy mass oscillates faster B. Light mass oscillates faster C. They oscillate at the same frequency.

12 Clicker question 3 I have a heavy mass and a light mass, each hanging on a similar spring. What can you say about the oscillation frequencies? A. Heavy mass oscillates faster B. Light mass oscillates faster C. They oscillate at the same frequency. If m goes down, ω goes up.

13 Clicker question 3 I have two equal masses on a stiff spring and a soft spring. What can you say about the oscillation frequencies? A. Stiff spring oscillates faster B. Soft spring oscillates faster C. They oscillate at the same frequency. Stiff spring is a large k. If k goes up, ω goes up.

14 Mass on a spring What does the motion of with look like? Cosine function oscillates between -1 and 1. A The factor A (amplitude) multiplies cosine and sets the maximum displacement that the oscillator reaches from the equilibrium point.

15 What does the motion of with look like? is the angular frequency (rad/s). It sets how quickly the system oscillates. The time it takes to increase by (a complete cycle) is the period The frequency is how many cycles are completed per second.. Frequency SI unit is hertz (Hz). Units of cycles/second or s -1 are also used. Note that.

16 What does the motion of with look like? The phase determines where along the cosine curve the oscillator starts (at t=0). For the oscillator starts at. For the oscillator starts at. For the oscillator starts at and moving in the x direction. For the oscillator starts at and moving in the +x direction.

17 Clicker question 4 The position of a mass on a spring as a function of time is shown below. At time P, which of the following statements about the velocity and acceleration is true? A. v>0 and a>0 B. v>0 and a<0 C. v<0 and a>0 D. v<0 and a<0 E. v>0 and a=0

18 Clicker question 4 The position of a mass on a spring as a function of time is shown below. At time P, which of the following statements about the velocity and acceleration is true? A. v>0 and a>0 B. v>0 and a<0 C. v<0 and a>0 D. v<0 and a<0 E. v>0 and a=0 Hint: Consider the displacement x at a slightly later time P+δP. As time increases, the object will move further along the path which increases x so the velocity must be positive. Since x is positive, force and acceleration must be negative

19 Clicker question 5 The solid curve is a graph of. The dotted curve is a graph of where φ is a phase constant whose magnitude is less than π/2. Is φ positive, negative or zero? A. φ<0 B. φ>0 C. φ=0 Hint: Remember that cosθ has a maximum at θ=0. We observe a maximum at ωt slightly greater than 0. The maximum value for cosθ is at θ = 0, 2π, 4π, 6π, So we need ωt+φ = 0, 2π, 4π, 6π, Since ωt is only slightly greater than 0 and φ <π/2 we need ωt+φ =0. Since ωt>0, φ<0.

20 Clicker question 5 The solid curve is a graph of. The dotted curve is a graph of where φ is a phase constant whose magnitude is less than π/2. Is φ positive, negative or zero? A. φ<0 B. φ>0 C. φ=0

Oscillatory Motion and Wave Motion

Oscillatory Motion and Wave Motion Oscillatory Motion Simple Harmonic Motion Wave Motion Waves Motion of an Object Attached to a Spring The Pendulum Transverse and Longitudinal Waves Sinusoidal Wave Function