Let's explore the relationship between the period and frequency of an object's motion to determine how changing one affects the other.
1. Understanding Period and Frequency:
- The period (T) is the time it takes for one complete cycle of motion.
- The frequency (f) is the number of cycles that occur per unit time and is the reciprocal of the period. Mathematically, [tex]\( f = \frac{1}{T} \)[/tex].
2. Original Period and Frequency:
- Let the original period be [tex]\( T \)[/tex].
- Hence, the original frequency would be [tex]\( f = \frac{1}{T} \)[/tex].
3. Tripling the Period:
- If the period is tripled, the new period becomes [tex]\( 3T \)[/tex].
4. New Frequency:
- The new frequency [tex]\( f' \)[/tex] with the new period [tex]\( 3T \)[/tex] will be [tex]\( f' = \frac{1}{3T} \)[/tex].
5. Comparing the New Frequency to the Original:
- The factor by which the new frequency is reduced compared to the original frequency is found by dividing the original frequency by the new period:
[tex]\[
f' = \frac{1}{3T} = \frac{1}{3} \times \frac{1}{T} = \frac{f}{3}
\][/tex]
6. Conclusion:
- Since the new frequency is [tex]\( \frac{f}{3} \)[/tex], it is one-third of the original frequency.
Thus, if the period of an object's motion is tripled, the frequency of its motion will be d. one-third as big.
