OSCILLATIONS (VIBRATIONS) SHM

Oscillation (Vibration) Vibration: back and forth or up and down motion that repeats itself at equal intervals of time on the same path. Examples: pendulum, spring & mass. Period T: time for a complete oscillation (s) Frequency f: the number of oscillations per unit of time. It is measured in Hz or s -1. One vibration per second is 1 hertz; two vibrations per second is 2 hertz, and so on. Higher frequencies are measured in kilohertz (kHz, thousands of hertz), and still higher frequencies in megahertz (MHz, millions of hertz) or gigahertz (GHz, billions of hertz).

Displacing a Spring results in Simple Harmonic Motion Releasing a strained a spring results in oscillating motion due to the spring restoring force This type of motion is called Simple Harmonic Motion

Simple Harmonic Motion

Period & Frequency If n oscillations are happening in the time t, then: T = t/n, and f = n/t. Therefore: T= 1/f, f = 1/T, or fT =1 Example: A church bell produces 20 “ding-dongs” in 10 s (1 ding- dong per oscillation). What is the period of vibration of the bell? T=10s/20 =0.5s What is the frequency of vibration of the bell? f=20/10s = 2Hz

Finding the spring constant k m: mass (kg) g = 9.8 m/s 2 x = stretch (m) Weight = Spring Force mg = kx k = mg/x

Period of The Spring and Mass System m: mass (kg), k spring (elastic) constant (N/m) Frequency: f = 1/T

Newton’s 2 nd Law & Ideal Springs Applying Newton’s 2 nd Law to a stretched ideal spring:  F = ma = -kx The acceleration of the spring is a = - (k/m). x The acceleration of the spring at any point in the motion is proportional to the displacement of the spring

The mass in a spring and mass system (k = 8N/m, m = 0.5 Kg) is at 5cm away from the equilibrium position. What is the acceleration of the mass? a= -kx/m a = -8 x 0.05/0.5 = m/s 2 What is the period of the system?

Vibration of a Pendulum The period of a pendulum increases with the length l. A long pendulum has a longer period than a short pendulum; that is, it swings to and fro less frequently than a short pendulum. A grandfather's clock pendulum with a length of about 1 m, for example, swings with a leisurely period of 2 s, while the much shorter pendulum of a cuckoo clock swings with a period that is less than a second. The period of a pendulum depends inversely to the acceleration of gravity g. Oil and mineral prospectors use very sensitive pendulums to detect slight differences in this acceleration. The acceleration due to gravity varies due to the variety of underlying formations.

Example T, f A mass attached to a string (pendulum) oscillates 20 times in 5s. What is the period and the frequency of the oscillation? T = 5s/20 = 0.25 s f = 20/5s = 4 Hz Check Tf = 1, 4 Hz x.25 s = 1 a) 4 sb) 0.25 s c) 100 s a) 4 Hzb) 0.25 Hz c) 100 Hz

EX. Simple Pendulum A geologist uses a pendulum of length 0.171m and counts 72 complete swings in a time 60 seconds. What is the value of g? a) 8.9 m/s 2 b) 9.7 m/s 2 c) 9.8 m/s 2 d)10 m/s 2 T = 60/72 = 0.83 s

Ex. Simple Pendulum On the moon g=1.6m/s 2 What is the period of the pendulum there? a) 1s b) 2s c) 3s d) 4s

A 1-meter-long pendulum has a bob with a mass of 1 kg. Suppose that the bob is now replaced with a different bob of mass 2 kg, how will the period of the pendulum change? A.It will double. B.It will halve. C.It will remain the same. D.There is not enough information. Vibrations of a Pendulum CHECK YOUR NEIGHBOR

A 1-meter-long pendulum has a bob with a mass of 1 kg. Suppose that the bob is now replaced with a different bob of mass 2 kg, how will the period of the pendulum change? A.It will double. B.It will halve. C.It will remain the same. D.There is not enough information. Vibrations of a Pendulum CHECK YOUR ANSWER Explanation: The period of a pendulum depends only on the length of the pendulum, not on the mass. So changing the mass will not change the period of the pendulum.

A 1-meter-long pendulum has a bob with a mass of 1 kg. Suppose that the bob is now tied to a different string so that the length of the pendulum is now 2 m. How will the period of the pendulum change? A.It will increase. B.It will decrease. C.It will remain the same. D.There is not enough information. Vibrations of a Pendulum CHECK YOUR NEIGHBOR

A 1-meter-long pendulum has a bob with a mass of 1 kg. Suppose that the bob is now tied to a different string so that the length of the pendulum is now 2 m. How will the period of the pendulum change? A.It will increase. B.It will decrease. C.It will remain the same. D.There is not enough information. Vibrations of a Pendulum CHECK YOUR ANSWER Explanation: The period of a pendulum increases with the length of the pendulum.

Simple Harmonic Motion and Uniform Circular Motion A ball is attached to the rim of a turntable of radius A The focus is on the shadow that the ball casts on the screen When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion

SHM and Circular Motion A spinning DVD is another example of periodic, repeating motion Observe a particle on the edge of the DVD The particle moves with a constant speed v c Its position is given by θ = ω t –ω = 2πf is the angular velocity Section 11.1

Simple Harmonic Motion Systems that oscillate in a sinusoidal matter are called simple harmonic oscillators –They exhibit simple harmonic motion –Abbreviated SHM The position can be described by y= A sin (2πƒt) –A is the amplitude of the motion The object moves back and forth between the positions y =  A –ƒ is the frequency of the motion Section 11.1

SHM: Velocity as a Function of Time Although the speed of the particle is constant, its y- component is not constant v y = v c cosθ = v c cos(ωt) = v c cos (2 π ƒ t) From the definition of velocity, we can express the particle’s speed as v = 2 π ƒ A cos (2 π ƒ t) = v = v max cos (2 π ƒ t), where v max = 2πfA Section 11.1

SHM graphically

Example The equation of and SHM is: x = 5cos20πt.Find: a) The frequency of the motion 2πf= 20π,f =10 Hz b) The period of the motion: T = 1/f = 0.1 s c) The amplitude: A = 5 m d) The maximum velocity v= ωA = 20πx5 = 100π m/s

ConcepTest 13.2 Speed and Acceleration 1) x = A 2) x > 0 but x < A 3) x = 0 4) x < 0 5) none of the above A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?

ConcepTest 13.2 Speed and Acceleration 1) x = A 2) x > 0 but x < A 3) x = 0 4) x < 0 5) none of the above A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously? If both v and a were zero at the same time, the mass would be at rest and stay at rest!NO pointva If both v and a were zero at the same time, the mass would be at rest and stay at rest! Thus, there is NO point at which both v and a are both zero at the same time. Follow-up: Where is acceleration a maximum?

ConcepTest 13.4 To the Center of the Earth A hole is drilled through the center of Earth and emerges on the other side. You jump into the hole. What happens to you ? 1) you fall to the center and stop 2) you go all the way through and continue off into space 3) you fall to the other side of Earth and then return 4) you won’t fall at all

gravity pulls you back toward the centerThis is Simple Harmonic Motion! You fall through the hole. When you reach the center, you keep going because of your inertia. When you reach the other side, gravity pulls you back toward the center. This is Simple Harmonic Motion! ConcepTest 13.4 To the Center of the Earth A hole is drilled through the center of Earth and emerges on the other side. You jump into the hole. What happens to you ? 1) you fall to the center and stop 2) you go all the way through and continue off into space 3) you fall to the other side of Earth and then return 4) you won’t fall at all Follow-up: Where is your acceleration zero?