Speed of a Wave in Steel Wire

In summary: Your calculation for F is now using the correct mass (5.0 kg) and your calculation for µ is using the correct formula. Therefore, the final calculation for the speed of the wave in the steel wire would be approximately 88.615 m/s.
  • #1
gmmstr827
86
1

Homework Statement



A 5.0-kg ball hangs from a steel wire 1.00 mm in diameter and 5.00 m long. What would be the speed of a wave in the steel wire?
Hint: Density of steel = þ_steel = 7.8 * 10^3 kg/m^3
Assume that the string is a cylinder to calculate the volume.

m = 5.0 kg
l (length) = 5.00 m
d = 1.00 mm = 0.001 m
r = 0.0005 m
þ_steel = 7.8 * 10^3 kg/m^3
g = 9.8 m/s^2

Homework Equations



V(volume)_cylinder = πlr^2
m = þV
F = mg
µ = m/l
v(velocity) = √(F/µ)

The Attempt at a Solution



V_cylinder = πlr^2 = π (5.00 m) (0.0005m)^2 ≈ 4.0 * 10^-6 m^3
m = þV = (7.8 * 10^3 kg/m^3) (4.0 * 10^-6 m^3) ≈ 0.0312 kg
F = mg = (0.0312 kg) (9.8 m/s^2) ≈ 0.306 N
µ = m/l = (5.0 kg)/(5.00 m) = 1 kg/m
v = √(F/µ) = √((0.306 N)/(1 kg/m)) = √(0.306) m/s ≈ 0.553 m/s

So, the speed of the wave in the steel wire would be approximately 0.553 m/s.

^^^ Does all of that look correct?
 
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  • #2
gmmstr827 said:

The Attempt at a Solution



V_cylinder = πlr^2 = π (5.00 m) (0.0005m)^2 ≈ 4.0 * 10^-6 m^3
m = þV = (7.8 * 10^3 kg/m^3) (4.0 * 10^-6 m^3) ≈ 0.0312 kg
F = mg = (0.0312 kg) (9.8 m/s^2) ≈ 0.306 N
µ = m/l = (5.0 kg)/(5.00 m) = 1 kg/m
v = √(F/µ) = √((0.306 N)/(1 kg/m)) = √(0.306) m/s ≈ 0.553 m/s

So, the speed of the wave in the steel wire would be approximately 0.553 m/s.

^^^ Does all of that look correct?

µ would be the mass per unit length of the wire, so you need to redo that part.

Also the F=mg is the force produced due to the 5 kg mass.
 
  • #3
rock.freak667 said:
µ would be the mass per unit length of the wire, so you need to redo that part.

Also the F=mg is the force produced due to the 5 kg mass.

Thanks for the help!

Ok, so now I have:

F = mg = (5.0 kg) (9.8 m/s^2) = 49 N
µ = m/l = (0.0312 kg)/(5.00 m) ≈ 0.00624 kg/m
v = √(F/µ) = √((49 N)/(0.00624 kg/m)) ≈ 88.615 m/s

Is THAT correct?
 
  • #4
That looks more correct.
 
  • #5


Yes, your calculations and equations are correct. However, it is important to note that the speed of a wave in a material is also dependent on the material's elasticity and stiffness, which can vary for different types of steel. Therefore, the calculated speed may not be an exact representation of the actual speed in the given steel wire. Additionally, it is always good to include units in your final answer to clarify the measurement being used.
 

What is the speed of a wave in steel wire?

The speed of a wave in steel wire is dependent on the physical properties of the wire, such as its density, elasticity, and tension. On average, the speed of a wave in steel wire is around 6000 meters per second.

How is the speed of a wave in steel wire calculated?

The speed of a wave in steel wire can be calculated using the formula v = √(T/μ), where v is the speed, T is the tension in the wire, and μ is the linear density (mass per unit length) of the wire. This formula is derived from the equation for the speed of a wave in a medium, which is v = √(F/μ), where F is the restoring force per unit length.

Does the speed of a wave in steel wire change with temperature?

Yes, the speed of a wave in steel wire does change with temperature. As the temperature increases, the wire's elasticity decreases, resulting in a decrease in the speed of the wave. This is due to the increase in thermal energy causing the atoms in the wire to vibrate more, making it more difficult for the wave to propagate through the wire.

How does the speed of a wave in steel wire compare to other materials?

The speed of a wave in steel wire is generally faster than in other materials, such as air or water. This is because steel is a much denser and more elastic material, allowing the wave to travel faster through it. However, the speed of a wave in steel wire is slower than in materials such as diamond or glass, which have higher elasticity and lower density.

How does the thickness of a steel wire affect the speed of a wave?

The thickness of a steel wire does not have a significant effect on the speed of a wave traveling through it. As long as the wire's other physical properties, such as density and tension, remain constant, the speed of the wave should remain consistent regardless of the wire's thickness. However, thicker wires may have a slightly slower speed due to increased internal friction and energy loss as the wave travels through the wire's larger cross-sectional area.

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