Speed of a Wave in Steel Wire

[gmmstr827]

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?

 

[rock.freak667]

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.

 

[gmmstr827]

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?

 

[rock.freak667]

That looks more correct.

 

[rwisz]

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.