How long it takes for the sound to travel through a piece of rod 2.5m long if it is made of copper? (the density of 8800kg / m3 and the modulus of elasticity 1.17x1011Pa).

To determine how long it takes for sound to travel through the copper rod, we first need to calculate the speed of sound in the copper material. The speed of sound in a solid material can be calculated using the following formula:

\[
v = \sqrt{\frac{E}{\rho}}
\]

Where:
- \( v \) is the speed of sound in the material,
- \( E \) is the modulus of elasticity (Pa),
- \( \rho \) is the density of the material (kg/m³).

Given:
- \( E = 1.17 \times 10^{11} \, \text{Pa} \),
- \( \rho = 8800 \, \text{kg/m}^3 \).

First, we compute the speed of sound \( v \):

\[
v = \sqrt{\frac{1.17 \times 10^{11}}{8800}}
\]

Next, the time \( t \) it takes for sound to travel through the rod can be calculated using the formula:

\[
t = \frac{L}{v}
\]

Where:
- \( L = 2.5 \, \text{m} \) (the length of the rod),
- \( v \) is the speed of sound we just calculated.

Let me perform the calculations.The speed of sound in the copper rod is approximately \( 3646.29 \, \text{m/s} \). 

The time it takes for sound to travel through the 2.5-meter-long copper rod is approximately \( 0.000686 \, \text{seconds} \), or \( 0.686 \, \text{milliseconds} \).

Would you like more details or have any further questions?

Here are five related questions:
1. How does the speed of sound in copper compare to that in air?
2. What is the effect of the rod's length on the travel time of sound?
3. How would changing the density of the material affect the speed of sound?
4. What is the speed of sound in other metals like aluminum or steel?
5. How do temperature and pressure affect the speed of sound in solids?

**Tip:** The modulus of elasticity plays a crucial role in determining how stiff a material is, and stiffer materials typically transmit sound faster.

This problem explores how to calculate the time it takes for sound to travel through a 2.5-meter copper rod, using concepts of wave propagation, density, and elasticity. We solve the problem step by step, applying key formulas: speed of sound (v = √(E/ρ)) and time (t = L/v). Suitable for Grades 10-12.

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