Question

(a) One end of a plastic foot ruler is held tightly at the edge of a table and the other end is plucked. Name the vibrations produced in the ruler.
(b) Now the ruler is pushed inside partially and plucked again from its free end. State with a reason whether the frequency of vibration increases or decreases.

Hint:

For vibrating strings or rulers, remember this simple rule: shorter length = higher frequency = higher pitch/shriller sound.

Answer 1

(a) Name the vibrations:
When the ruler is plucked and then allowed to vibrate on its own, it undergoes free vibrations. However, in reality, forces like air resistance and internal friction in the material will cause the amplitude of these vibrations to gradually decrease. Therefore, the vibrations produced are more accurately called damped vibrations.
(b) Change in frequency:
Statement: The frequency of vibration increases.
Reason: The frequency of vibration of a cantilever (like a ruler held at one end) is inversely related to the square of its vibrating length (\(L\)), i.e., \(f \propto 1/L^2\). When the ruler is pushed inside, its effective vibrating length (\(L\)) decreases. Since the frequency is inversely proportional to the length squared, a shorter length results in a significantly higher frequency of vibration. A higher frequency corresponds to a shriller sound.