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Mechanical Vibrations: Free undamped single-degree-of-freedom systems · Natural Frequency of a Spring-Mass System

How mass, stiffness, gravity and length set the natural frequency of springs and pendulums vibrating freely.

Mechanical Vibrations · 23 questions

Try 3 questions

1. For a mass m on a spring of stiffness k, what is the undamped natural frequency in radians per second?

  1. Stiffness k divided by mass m
  2. The square root of k divided by m
  3. The square root of m divided by k
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Answer: The square root of k divided by m

Newton's second law gives m·x'' + k·x = 0, whose solution oscillates at ωn = √(k/m): stiffer springs speed it up, heavier masses slow it down.

2. You double the mass on a spring and keep the spring the same. What happens to the natural frequency?

  1. It stays the same
  2. It falls to about 71% of before
  3. It halves
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Answer: It falls to about 71% of before

Frequency scales with 1/√m, so doubling the mass divides it by √2, leaving about 0.71 of the original.

3. A spring is replaced with one four times stiffer, same mass. How does the natural frequency change?

  1. It doubles
  2. It rises by about half
  3. It becomes four times higher
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Answer: It doubles

Frequency grows with the square root of stiffness, and √4 = 2, so a fourfold stiffer spring only doubles the frequency.

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