OSCILLATIONS VIBRATIONS SHM Oscillation Vibration Vibration back and
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 (k. Hz, 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 f. T =1 Example: A church bell produces 20 “ding-dongs” in 10 s (1 dingdong per oscillation). What is the period of vibration of the bell? T=10 s/20 =0. 5 s What is the frequency of vibration of the bell? f=20/10 s = 2 Hz
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: SF = 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 = 8 N/m, m = 0. 5 Kg) is at 5 cm away from the equilibrium position. What is the acceleration of the mass? a= -kx/m a = -8 x 0. 05/0. 5 = - 0. 8 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 5 s. What is the period and the frequency of the oscillation? a) 4 s b) 0. 25 s c) 100 s T = 5 s/20 = 0. 25 s a) 4 Hz b) 0. 25 Hz c) 100 Hz f = 20/5 s = 4 Hz Check Tf = 1, 4 Hz x. 25 s = 1
EX. Simple Pendulum • A geologist uses a pendulum of length 0. 171 m 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. 6 m/s 2 What is the period of the pendulum there? a) 1 s b) 2 s c) 3 s d) 4 s
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. B. C. D. It will double. It will halve. It will remain the same. There is not enough information.
Vibrations of a Pendulum CHECK YOUR ANSWER 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. B. C. D. It will double. It will halve. It will remain the same. There is not enough information. 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.
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. B. C. D. It will increase. It will decrease. It will remain the same. There is not enough information.
Vibrations of a Pendulum CHECK YOUR ANSWER 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. B. C. D. It will increase. It will decrease. It will remain the same. There is not enough information. 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 vc • Its position is given by θ=ωt – 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 ycomponent is not constant • vy = vc cosθ = vc cos(ωt) = vc 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) = vmax cos (2 π ƒ t), where vmax = 2πf. A Section 11. 1
SHM graphically
Example • The equation of and SHM is: x = 5 cos 20πt. 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 = 20πx 5 = 100π m/s
Concep. Test 13. 2 Speed and Acceleration A mass on a spring in SHM has 1) x = A amplitude A and period T. At 2) x > 0 but x < A what point in the motion is v = 0 3) x = 0 and a = 0 simultaneously? 4) x < 0 5) none of the above
Concep. Test 13. 2 Speed and Acceleration A mass on a spring in SHM has 1) x = A amplitude A and period T. At 2) x > 0 but x < A what point in the motion is v = 0 3) x = 0 and a = 0 simultaneously? 4) x < 0 5) none of the above 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?
Concep. Test 13. 4 To the Center of the Earth A hole is drilled through the 1) you fall to the center and stop center of Earth and emerges on the other side. You jump into the hole. What happens 2) you go all the way through and continue off into space to you ? 3) you fall to the other side of Earth and then return 4) you won’t fall at all
Concep. Test 13. 4 To the Center of the Earth A hole is drilled through the 1) you fall to the center and stop center of Earth and emerges on the other side. You jump into the hole. What happens 2) you go all the way through and continue off into space to you ? 3) you fall to the other side of Earth and then return 4) you won’t fall at all 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! Follow-up: Where is your acceleration zero?
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