a piano string has a length of 2m and a density of 8,000kg/m3. when the tension in the string produces a strain of 1%, the fundamental note obtained is 170 Hz. calculate the modulus of elasticity for the material of the string.
This is not the formal solutiin though process is correct. Squrt should not be written it doesn't look good. Finally thanks for answer. I have done this but question bank answer was not matching. There may by mistake in qb.
A taut sonometer wire, 76 cm long, with a tension of 40 N is connected to a source of
alternating current. Two horse-shoe magnets of opposite poles are placed either side of the wire at its midpoint, and as current is allowed to pass through the sonometer, the resulting
forces set the wire in resonant vibration. If the density of the material of the wire is 8800 kg/m
3 and the wire is 1mm thick, obtain the frequency of the alternating current.
Why did the piano string go to therapy? Because it was under a lot of tension!
Now, let's calculate the modulus of elasticity for the material of the string using the given information. The modulus of elasticity (E) can be calculated using the formula:
E = (stress / strain)
First, let's calculate the strain using the formula:
strain = (change in length / original length)
Since the strain is given as 1%, we can write it as 0.01. Now, let's find the change in length using the formula:
change in length = strain * original length
change in length = 0.01 * 2m
change in length = 0.02m
Now, we can calculate the stress using the formula:
stress = (force / area)
Since the density (ρ) is given as 8,000 kg/m^3, we can calculate the mass (m) of the string as:
mass = density * volume
mass = 8,000 kg/m^3 * (2m * A) [since length x area = volume]
Given that the tension in the string produces a strain of 1%, we can also calculate the force (F) as:
force = tension * area
Now, let's substitute the formulas to get:
mass = 8,000 kg/m^3 * (2m * A)
force = tension * area
Using the formula for density and the given strain, we can substitute the mass and force equations into the stress formula:
stress = (force / area)
stress = [(tension * area) / area]
stress = tension
Thus, the stress is equal to the tension. So, we can rewrite the formula for the modulus of elasticity (E) as:
E = (tension / strain)
Now, we can substitute the given values to find E:
E = (tension / strain)
170 Hz = (tension / 0.02m)
Now let's solve for the tension:
tension = 170 Hz * 0.02m
tension = 3.4 N
Finally, we substitute the calculated tension back into the formula for the modulus of elasticity:
E = (tension / strain)
E = 3.4 N / 0.01
E = 340 N/m²
So, the modulus of elasticity for the material of the piano string is 340 N/m². I hope this helps, and remember, always keep a sense of humor while dealing with elastic moduli!
To calculate the modulus of elasticity for the material of the string, we can use the formula:
Modulus of Elasticity (E) = (Tension / Strain) * (Length / Area)
Given values:
Length of the string (L) = 2m
Density of the string (ρ) = 8,000 kg/m^3
Strain (ε) = 1% = 0.01 (in decimal form)
Fundamental frequency (f) = 170 Hz
Step 1: Calculate the mass per unit length of the string.
The mass per unit length of the string can be calculated using the formula:
Mass per unit length (μ) = Density * Area
Since the string is assumed to be linear, the cross-sectional area does not change along its length. Therefore, we can assume a constant cross-sectional area for simplicity.
Step 2: Calculate the tension in the string (T).
The tension in the string can be calculated using the formula:
Tension (T) = mass per unit length * Length * (2πf)^2
Step 3: Calculate the modulus of elasticity (E).
Modulus of Elasticity (E) = Tension / (Strain * (Length / Area))
Now let's calculate each step:
Step 1:
μ = ρ * Area
Step 2:
T = μ * L * (2πf)^2
Step 3:
E = T / (ε * (L / Area))
You will need to provide the value of the cross-sectional area of the string in order to complete the calculations. Once you have the value, plug it into the formulas and calculate the modulus of elasticity (E).
f = sqrt (T/[m/L]) / 2L
m = 8000 A L
so
170 = sqrt (T/8000A)/4
680 = sqrt (T/8000A)
also
1% = .01 = T/ (E A)
so
T/A = .01 E
so
680 = sqrt (.01 E/8000)
462400 = .01 E/8000
E = 3.7 *10^11