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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Athin 4.5m metal od sustains a longitudinal standing wave with vibration atinodes at exch endof the od Thre ae no cther antinodes. The density and Young's modulus of this metal ar. respectively, 8300 kgm nd 7.11010. Whats the frequency the rod vibration? Number [| Units

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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Uploaded ImageAthin 4.5m metal od sustains a longitudinal standing wave with vibration atinodes at exch endof the od Thre ae no cther antinodes. The density and Young's modulus of this metal ar. respectively, 8300 kgm nd 7.11010. Whats the frequency the rod vibration? Number [| Units

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Asked by: sai
Created at: 2025-06-08 04:10:09

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**Given Problem:** A thin 4.5-m metal rod sustains a longitudinal standing wave with vibration antinodes at each end of the rod. There are no other antinodes. The density and Young's modulus of this metal are, respectively, 8800 kg/m³ and 7.1×10¹⁰ N/m². What is the frequency of the rod's vibration? --- ### **Definition of Parameters** - **Length of rod, \( L \) = 4.5 m** - **Density, \( \rho \) = 8800 kg/m³** - **Young's Modulus, \( Y \) = 7.1 \times 10^{10} \) N/m²** - **Standing wave with antinodes at both ends and no other antinodes means the rod vibrates in its fundamental mode.** --- ### **Step 1: Find the Speed of Sound in the Rod** The speed of longitudinal waves in a rod is given by: \[ v = \sqrt{\frac{Y}{\rho}} \] **Substitute the given values:** \[ v = \sqrt{\frac{7.1 \times 10^{10}}{8800}} \] \[ v = \sqrt{8.068 \times 10^6} \] \[ v \approx 2,840 \text{ m/s} \] **Explanation:** The speed of sound in a solid rod depends on the material's Young's modulus and its density. The formula calculates how fast vibrations travel along the rod. --- ### **Step 2: Find the Fundamental Frequency** For a rod with antinodes at both ends (fundamental mode), the length of the rod corresponds to half of the wavelength: \[ L = \frac{\lambda}{2} \implies \lambda = 2L \] The frequency is given by: \[ f = \frac{v}{\lambda} \] Substitute values: \[ \lambda = 2 \times 4.5 = 9.0 \text{ m} \] \[ f = \frac{2840}{9.0} \] \[ f \approx 316 \text{ Hz} \] **Explanation:** The fundamental frequency for a rod with antinodes at both ends is determined by the wave speed divided by twice the rod's length (since the wavelength is twice the length for this mode). --- ### **Final Answer Summary** **The frequency of the rod's vibration is approximately 316 Hz.**

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