Vibrations and Waves

When a vibration or an oscillation repeats itself, back and forth, over the same path, the motion is periodic. The simplest examples: pendulum or oscillating of a coil spring.

Restoring Force Important: the restoring force F is not constant, but varies with position!

At what points of trajectory does the restoring force have maximum At what points of trajectory does the restoring force have maximum? Zero?

Some Terms: displacement (x) : the distance x from equilibrium point at any moment; amplitude (A): the maximum displacement; one cycle refers to the complete go-and-fro motion from some initial point back to the same point; period (T): time, required for one complete cycle; frequency (f): the number of complete cycles per second.

At what points of trajectory does the displacement of object have maximum? Zero? At what points of trajectory does the velocity of an object have maximum? Zero?

What is the mutual direction of the displacement and acceleration while oscillatory motion?

Energy in the Simple Harmonic Oscillator

The total mechanical energy of a simple harmonic oscillator is proportional to the square of the amplitude.

The Period of Oscillations

Important: The period of simple oscillations doesn’t depend on the amplitude!!

The Simple Pendulum

The period doesn’t depend on the mass of bob, it depends only on the length of the thread.

Damped Harmonic Motion

Forced Vibrations; Resonance

The increase of amplitude near f=f0 is known as resonance.

At what frequencies (if any) except for f0 might be observed a resonance?

You have a grandfather clock (with a pendulum) that keeps perfect time on Earth. If you were to transport this clock to the Moon, would its period of oscillations increase, decrease, or stay the same? Explain. A mass is oscillating up and down on a vertical spring. When the mass is below the equilibrium point and moving down, what direction is its acceleration? Is the mass speeding up or slowing down?

How does the natural frequency of a swing change when you move from sitting down to standing up? A man enters a tall tower, needing to know its height. He has a long rope and a heavy piece of iron. He also has a wristwatch. How can he estimate the height of tower?

You hold one end of a spring in your hand and hang a block from the other end. After lifting the block up slightly and releasing it, you find that it oscillates up and down at a frequency of 2 Hertz. At which of the following frequencies could you jiggle your hand up and down and produce resonance: 5 hertz, 4 hertz, 1.5 hertz, 1 hertz, or 0.5 hertz? You stand to the side of the low point of a child’s swing and always push the child in the same direction. Which of the following multiplies of the fundamental frequency will not produce resonance: 1/3; ½; 1, or 2?