A sinusoidal wave travels along a string. The time for a particular point to move from maximum placement to zero is 0.135s. What are the (a) period and (b) frequency? (c) The wavelength is 1. 40m, what is the wave speed?

Answer

To solve the problem step by step, we will break it down into three parts as per the question.

### Step 1: Find the Period (T)

Given that the time for a particular point to move from maximum displacement to zero is 0.135 seconds, we can determine the period of the wave.

1. The motion from maximum displacement to zero represents a quarter of the wave's period (T/4).
2. Therefore, we can express the relationship as:
\[ \frac{T}{4} = 0.135 \text{ seconds} \]
3. To find the full period (T), we multiply both sides by 4:
\[ T = 0.135 \times 4 = 0.54 \text{ seconds} \]

### Step 2: Find the Frequency (f)

The frequency is the reciprocal of the period.

1. The formula for frequency is:
\[ f = \frac{1}{T} \]
2. Substituting the value of T we found:
\[ f = \frac{1}{0.54} \approx 1.85 \text{ Hz} \]

### Step 3: Find the Wave Speed (v)

We are given the wavelength (λ) as 1.40 m. The wave speed can be calculated using the formula:
\[ v = f \times \lambda \]

1. Substituting the values we have:
\[ v = 1.85 \text{ Hz} \times 1.40 \text{ m} \]
2. Calculating the wave speed:
\[ v \approx 2.59 \text{ m/s} \]

### Final Answers:
(a) The period (T) is 0.54 seconds.
(b) The frequency (f) is approximately 1.85 Hz.
(c) The wave speed (v) is approximately 2.59 m/s.

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