Question 3 of 23beginner🔧 ApplyNumerical2 marks

A steel wire 0.72 m long has a mass of 5.0×10−35.0 \times 10^{-3} kg. If the wire is under a tension of 60 N, what is the speed of transverse waves on the wire?

Correct Answer

Given: Length of wire, L = 0.72 m; Mass of wire, m = 5.0×105.0 \times 10⁻³ kg; Tension, T = 60 N

Step 1: Calculate linear mass density (μ) μ = m/L = 5.0×105.0 \times 10⁻³ kg / 0.72 m = 6.9×106.9 \times 10⁻³ kg m⁻¹

Step 2: Apply the wave speed formula The speed of transverse waves on a stretched string is given by: v = T/μ\sqrt{T/μ}

Step 3: Substitute values and calculate v = \sqrt{60 N / $$6.9 \times 10$$⁻³ kg m⁻¹} = 93 m s⁻¹

Answer: The speed of transverse waves on the wire is 93 m s⁻¹.

Exercise: EXAMPLES | Q: Example 14.3 | (Page 8)
For More Understanding

Explanation

This numerical problem from Example 14.3 tests the understanding of wave speed on a stretched string. The speed depends only on the tension (T) and the linear mass density (μ) of the string, as stated in the textbook context. The formula v = T/μ\sqrt{T/μ} is derived from dimensional analysis and Newton's Laws of Motion. The linear mass density (μ) is the mass per unit length, calculated by dividing the total mass by the length of the wire. This is a straightforward application of the wave speed formula for transverse waves on a string.

Solution Steps

  1. Step 1: Calculate linear mass density μ = m/L = 5.0×105.0 \times 10⁻³ kg / 0.72 m = 6.9×106.9 \times 10⁻³ kg m⁻¹

  2. Step 2: Apply the wave speed formula v = T/μ\sqrt{T/μ}

  3. Step 3: Substitute values: v = \sqrt{60 N / $$6.9 \times 10$$⁻³ kg m⁻¹} = 93 m s⁻¹

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