Chapter 14 Vibrations and Waves 14 1 Periodic

Chapter 14 – Vibrations and Waves

14. 1 Periodic Motion Every swing follows the same path This action is an example of vibrational motion - mechanical oscillations around an equilibrium point

14. 1 Periodic Motion Each trip back and forth takes the same amount of time This motion, which repeats in a regular cycle, is an example of periodic motion

14. 1 Periodic Motion The simplest form of periodic motion can be represented by a mass oscillating on the end of a coil spring. - mounted horizontally - ignore mass of spring - no friction

14. 1 Periodic Motion - Any spring has a natural length at which it exerts no force on the mass, m. This is the equilibrium position - Moving the mass compresses or stretches the spring, and the spring then exerts a force on the mass in the direction of the equilibrium position - This is the restoring force

14. 1 Periodic Motion - At the equilibrium position x = 0 and F = 0 - The further the mass is moved (in either direction) from the equilibrium position, the greater the restoring force, F - The restoring force is directly proportional to the displacement from the equilibrium position

14. 1 Periodic Motion Hooke’s Law (restoring force of an ideal spring) F = -kx - k is the “spring constant” (units of N/m) - a stiffer spring has a larger value of k (more force is required to stretch it) - The minus sign indicates the restoring force is always opposite the direction of the displacement - Note, the force changes as x changes, so the acceleration of the mass is not constant

14. 1 Periodic Motion A spring stretches by 18 cm when a bag of potatoes with a mass of 5. 71 kg is suspended from its end. a) Determine the spring constant.

14. 1 Periodic Motion A 57. 1 kg cyclist sits on a bicycle seat and compresses the two springs that support it. The spring constant equals 2. 2 x 104 N/m for each spring. How much is each spring compressed? (Assume each spring bears half the weight of the cyclist)

14. 1 Periodic Motion - The spring has the potential to do work on the ball - The work however, is NOT W = Fx, because F varies with displacement - We can use the average force: 1 1 __ __ F = (0 + kx) = kx 2 2 1 1 __ __ W = Fx = kx(x) = kx 2 2 2

14. 1 Periodic Motion Potential Energy in a Spring 1 __ PEsp = kx 2 2 The potential energy in a spring is equal to one-half times the product of the spring constant and the square of the displacement

14. 1 Periodic Motion 2. 0 cm A 0. 5 kg block is used to compress a spring with a spring constant of 80. 0 N/m a distance of 2. 0 cm (. 02 m). After the spring is released, what is the final speed of the block?

14. 1 Periodic Motion - First, the object is stretched from the equilibrium position a distance x = A - The spring exerts a force to pull towards equilibrium position - Because the mass has been accelerated, it passes by the equilibrium position with considerable speed - At the equilibrium position, F = 0, but the speed is a maximum

14. 1 Periodic Motion - As its momentum carries it to the left, the restoring force now acts to slow (decelerate) the mass, until is stops at x = -A - The mass then begins to move in the opposite direction, until it reaches x = A The cycle then repeats (periodic motion)

14. 1 Periodic Motion Terms for discussing periodic motion Displacement – the distance of the mass from the equilibrium point at any moment Amplitude – the greatest distance from the equilibrium point

14. 1 Periodic Motion Cycle – a complete to-and-fro motion (i. e. – from A to –A and back to A) Period (T) – the time required to complete one cycle Frequency (f) – the number of cycles per second (measured in Hertz, Hz)

14. 1 Periodic Motion Frequency and Period are inversely related If the frequency is 5 cycles per second (f = 5 Hz), what is the period (seconds per cycle)? 1 __ s 5 1 __ f= T 1 __ and T = f

14. 1 Periodic Motion Any vibrating system for which the restoring force is directly proportional to the negative of the displacement is said to exhibit simple harmonic motion.

14. 1 Periodic Motion Period of a Mass-Spring System

14. 1 Periodic Motion A 1. 0 kg mass attached to one end of a spring completes one oscillation every 2. 0 s. Find the spring constant What size mass will make the spring vibrate once every 1. 0 s?

14. 1 Periodic Motion Simple harmonic motion can also be demonstrated with a simple pendulum The net force on the pendulum is a restoring force x = min v= max a = min x = max v = min a = max

14. 1 Periodic Motion

14. 1 Periodic Motion

14. 1 Periodic Motion

14. 1 Periodic Motion

14. 1 Periodic Motion

14. 1 Periodic Motion Period of a Pendulum The period of a pendulum is equal to two pi times the square root of the length of the pendulum divided by the acceleration due to gravity

14. 1 Periodic Motion What is the period of a 99. 4 cm long pendulum? What is the period of a 3. 98 m long pendulum?

14. 1 Periodic Motion A desktop pendulum swings back and forth once every second. How tall is this pendulum?

14. 1 Periodic Motion Resonance The condition in which a time dependent force can transmit large amounts of energy to an oscillating object leading to a larger amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.