Chapter 14 – Vibrations and Waves

Every swing follows the same path This action is an example of vibrational motion 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 - The minus sign indicates the restoring force is always opposite the direction of the displacement - 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) - 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 10 4 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: F = 1 __ 2 (0 + kx) = __ 2 1 kx W =Fx = __ 2 1 kx(x) = __ 2 1 kx 2

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

14.1 Periodic Motion 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? 2.0 cm

- 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 __ 5 s f = 1 __ T and T = 1 __ 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 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 = max v = min a = max x = min v= max a = min

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.