Final answer:
Using the given length of the steel rod and the time for sound to travel through it, we calculated the speed of sound. Then, applying the relationship between speed, density, and Young's modulus for longitudinal waves, we found the density of the steel to be approximately 7,800 kg/m³, which is closest to option B) 7,826 kg/m³.
Step-by-step explanation:
To find the density of steel, we need to use the speed of sound in the steel rod and the modulus of elasticity (also known as the Young's modulus) of steel. The speed of sound v can be calculated by dividing the length L of the steel rod by the time t it takes for the sound to travel through it:
v = L / t
Given the length L = 3m and time t = 610-4s, we get the speed of sound in the steel rod:
v = 3m / 610-4s = 5000m/s (speed of sound in steel)
The relationship between the speed of sound v, the density of steel ρ, and Young's modulus E for a longitudinal wave is:
v = √(E / ρ)
Therefore, the density ρ can be calculated from the speed of sound v and Young's modulus E:
ρ = E / v2
Given that Young's modulus E = 19.51010Pa and v = 5000m/s:
ρ = 19.51010Pa / (5000m/s)2
ρ = 7,800kg/m3 (to three significant figures)
Therefore, the answer is B) 7,826 kg/m3.