Section 1: Simple Harmonic motion Chapter 11 Section 1: Simple Harmonic motion

Objectives Identify the conditions of simple harmonic motion. Explain how force, velocity, and acceleration change as an object vibrates with simple harmonic motion. Calculate the spring force using Hooke’s law.

Hooke’s Law Hooke's law is a principle of physics that states that the force needed to extend or compress a spring by some distance x is proportional to that distance. That is F=-kx, where k is a constant factor characteristic of the spring, its stiffness. Hooke's equation in fact holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, a musician plucking a string of a violin, or the filling of a party balloon. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.

Hooke’s Law Hooke's law is named after the 17th century British physicist Robert Hooke. He first stated this law in 1660 as a Latin anagram,[1][2] whose solution he published in 1678 as Ut tensio, sic vis; literally translated as: "As the extension, so the force" or a more common meaning is "The extension is proportional to the force".

Periodic motion A periodic motion -is a repeated motion such as that of an acrobat swinging on a trapeze. Another examples can be a child playing on a swing. One of the simplest types of back –and forth periodic motion is the motion of a mass attached to a spring

Periodic motion The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0). At equilibrium: The spring force and the mass’s acceleration become zero. The speed reaches a maximum. At maximum displacement: The spring force and the mass’s acceleration reach a maximum. The speed becomes zero.

Periodic motion In an ideal system, the mass spring system would oscillate indefinitely. But in the physical world , friction retards the motion of the vibrating mass, and the mass-spring system eventually comes to rest. This effect is called damping.

spring force = –(spring constant  displacement) Hooke’s law Measurements show that the spring force, or restoring force, is directly proportional to the displacement of the mass. This relationship is known as Hooke’s Law: Felastic = –kx spring force = –(spring constant  displacement) The quantity k is a positive constant called the spring constant.

Simple harmonic motion Describes any periodic motion that is the result of a restoring force is proportional to displacement. Examples of simple harmonic consists of objects that have a back and forth motion over the same path.

Hooke’s law example a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant? Solution: Define Given: m = 0.55 kg x = –2.0 cm = –0.20 m g = 9.81 m/s2

solution Unknown: k = ? Diagram:

solution Plan: Fnet = 0 = Felastic + Fg Felastic = –kx Fg = –mg –kx – mg = 0

3. Calculate

Student guided practice Do problems 2 and 3

Simple pendulum A simple pendulum consists of a mass called a bob, which is attached to a fixed string. The forces acting on the bob at any point are the force exerted by the string and the gravitational force.

Simple pendulum At any displacement from equilibrium, the weight of the bob (Fg) can be resolved into two components. The x component (Fg,x = Fg sin ) is the only force acting on the bob in the direction of its motion and thus is the restoring force. The magnitude of the restoring force (Fg,x = Fg sin ) is proportional to sin . When the maximum angle of displacement q is relatively small (<15°), sin  is approximately equal to  in radians. As a result, the restoring force is very nearly proportional to the displacement. Thus, the pendulum’s motion is an excellent approximation of simple harmonic motion.

Simple harmonic motion

Homework Do Hooke's worksheet problems 1 to 5

Video Let’s look at a video

Closure Today we learned about hooke’s law Next class we re going to continue with simple harmonics.