Section 2 review
Most systems have a natural frequency at which they oscillate when disturbed; many have several such natural frequencies. Systems may exhibit an especially large amplitude of motion at their natural frequency. Resonance describes the large-amplitude response that occurs when an oscillatory system is driven with a force that matches its natural frequency. The natural frequency of a simple pendulum depends on the acceleration due to gravity (g) and the length of the pendulum but not upon the mass of the bob or the amplitude. The natural frequency of a mass-and-spring assembly depends upon the spring constant and the mass of the attached object but not upon g or the amplitude. Read the text aloud
periodic force, resonance

T=2π L g
f= 1 2π k m

Review problems and questions

  1. Describe the relationship among resonance, amplitude, force, and natural frequency. Read the text aloud Show
  1. Josué and Dalia are building a small pendulum to serve as a primitive stopwatch for a ball-and-ramp experiment. They hang a dense metal nut from a lab stand with string. They discover that their pendulum has a period T of ½ second.
    1. How long is the pendulum?
    2. Suppose that the partners now want their pendulum to swing back and forth once every second, not every half-second. Josué argues that they should double the length of the string. Dalia argues that the string should only be 50% longer. Which partner (if either) is right? Read the text aloud Show
  1. Tracie and Bob are performing a series of experiments with small, dense metal weights hanging from springs (which are attached to sturdy lab stands). They find a mass-and-spring combination that bobs up and down once every 2 s, after being gently stretched and then released. Their teacher asks them to double the system’s frequency. Which of the following changes might achieve this?
    1. Replace the spring with a stiffer (harder-to-stretch) spring.
    2. Replace the metal weight with a lighter (less massive) object.
    3. Either a or b will increase the frequency.
    4. Neither a nor b will increase the frequency. Read the text aloud Show
  1. A 1.0 kg mass hangs on a spring with a spring constant of k = 158 N/m. The mass is pulled down 10 cm and released, causing the system to move in simple harmonic motion with a natural frequency of 2.0 Hz and an amplitude of 10 cm. What is the new frequency if the mass, spring constant, and amplitude are changed as described below?
    1. The mass and spring constant are doubled, and the amplitude is unchanged.
    2. The mass is quadrupled, and the spring constant and amplitude are unchanged.
    3. The spring constant is quadrupled, and the mass and amplitude are unchanged.
    4. The amplitude is doubled, and the mass and spring constant are unchanged. Read the text aloud Show

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