Any regular vibrations or oscillations that repeat the same movement on either side of the equilibrium position and are a result of a restoring force Simple Harmonic Motion Results in back and forth motion over the same path Restoring force: a force that tries to return an object to equilibrium (center resting position).

Amplitude Pendulum: measured by the angle between the pendulum’s equilibrium position and its maximum displacement. Mass-spring system: maximum amount the spring is stretched or compressed from its equilibrium position. Units: radian (rad) or degrees and the meter (m). Maximum displacement from equilibrium

Period Time it takes to complete one cycle Units: Seconds Frequency Number of cycles or vibrations per unit of time Units: hertz (Hz) Period & Frequency are inversely related

Period vs. Amplitude

Period vs. Mass

Period vs. Length

Period squared vs. Length

Period of a Simple Pendulum in SHM The period of a simple pendulum depends on the length and on the free-fall acceleration. The period does not depend on the mass of the bob or on the amplitude (for small angles).

Kinetic Energy energy of an object due to the object’s motion depends on speed and mass.

Potential Energy Stored energy associated with an object because of the position, shape, or condition of the object. Gravitational potential energy is the energy an object has because of its position in a gravitational field. GPE depends on height from a zero level and the mass of the object. PE g = mgh gravitational PE = mass  free-fall acceleration  height

Potential Energy, continued Elastic potential energy is the energy available for use when a deformed elastic object returns to its original configuration. The symbol k is called the spring constant, a parameter that measures the spring’s resistance to being compressed or stretched.

Mechanical Energy The sum of kinetic energy and all forms of potential energy associated with an object or group of objects ME = KE + ∑PE Is often conserved ME i = ME f initial mechanical energy = final mechanical energy (in the absence of friction)

Energy in pendulums & springs

SHM in springs The direction of the force acting on the mass (F elastic ) is opposite the direction of the mass’s displacement from equilibrium (x = 0).

SHM in hanging spring

SHM in springs At equilibrium: Spring force? Speed? Kinetic Energy? Elastic Potential Energy? At maximum displacement: Spring force? Speed? Kinetic Energy? Elastic Potential Energy?

Hooke’s Law The spring force, or restoring force, is directly proportional to the displacement of the mass. This relationship is known as Hooke’s Law: F elastic = kx spring force = spring constant  displacement

Practice Questions 1.If 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? 2.Suppose the spring from above is replaced with a spring that stretches 36 cm from its equilibrium position. What is the spring constant? Is this spring stiffer or less stiff? 3.How much force is required to pull a spring 3.0 cm from its equilibrium position if the spring constant is 2700 N/m?

Period of a Mass-Spring System The period of an ideal mass-spring system depends on the mass and on the spring constant. The period does not depend on the amplitude.

Practice: Mass-Spring System 1)A 125 N object vibrates with a period of 3.5 s when hanging from a spring. What is the spring constant of the spring? 2)A spring of 30.0 N/m is attached to different masses, and the system is set in motion. Find the period and frequency of vibration for masses of 2.3 kg. 3)A child’s toy that is made to shoot ping pong balls consists of a tube, a spring (k = 18 N/m) and a catch for the spring that can be released to shoot the balls. When a ball is loaded into the tube, it compresses the spring 9.5 cm. If you shoot a ping pong ball straight out of this toy, what is its maximum speed?

Section Review Which of these periodic motions are simple harmonic? –a child swinging on a playground swing (θ = 45) –a CD rotating in a player –an oscillating clock pendulum (θ = 10) A pinball machine uses a spring that is compressed 4.0 cm to launch a ball. If the spring is 13 N/m, what is the force on the ball? How does the restoring force acting on a pendulum bob change as the bob swings toward the equilibrium position? How do the bob’s acceleration (along the direction of motion) and velocity change?

Section Review A child swings on a playground swing with a 2.5 m long chain. –What is the period and frequency of the child in motion A 0.75 kg mass attached to a vertical spring stretches the spring 0.30 m. –What is the spring constant?