Understanding the Relationship between Bulk Modulus Equations

[Zorodius]

I'm sorry if this should be obvious to me, but it's not.

My book provides this equation for the bulk modulus:

Equation 1:
$$B = -\frac{\Delta p}{\Delta V / V}$$

It says, by way of explanation, that it follows from a relationship they gave earlier:

Equation 2:
$$p = B \frac{\Delta V}{V}$$

(where p is the pressure, B is the bulk modulus, and V represents volume.)

However, Equation 1 is easily rearranged into:

Equation 3:
$$B \frac{\Delta V}{V} = - \Delta p$$

Comparing equation 2 and 3 suggests:

Equation 4:
$$p = - \Delta p$$

Which is, as far as I can tell, complete nonsense.

Are equation 1 and 2 really equivalent statements? If so, how?

 

[Doc Al]

Equation 1:
$$B = -\frac{\Delta p}{\Delta V / V}$$

This is the definition of the Bulk modulus. (The minus sign is just to make B positive: an increase in pressure usually creates a decrease in volume.)

It says, by way of explanation, that it follows from a relationship they gave earlier:

Equation 2:
$$p = B \frac{\Delta V}{V}$$

This makes no sense to me. As you point out, it seems they are confusing pressure with change in pressure. What book is this?

 

[Zorodius]

Thanks a lot for the reply, I'm glad to know I'm not totally crazy yet.

This makes no sense to me. As you point out, it seems they are confusing pressure with change in pressure. What book is this?

Halliday, Resnick, and Walker, "Fundamentals of Physics", extended sixth edition. The erroneous definition of bulk modulus appears on page 286, in chapter 13.

Between this and other gems like "all friction is caused by cold welding" and "an element of a string oscillating in a transverse wave has maximum elastic potential energy at zero displacement" (both paraphrased, but exactly what the book claims), I'm about ready to conduct an experiment in inelastic collisions between this book and the garbage can

 

[maverick280857]

Have you been able to locate an authentic errata which confirms this? Otherwise, the publishers should know about this. I suggest you refer to the older physics book by Resnick and Halliday (published sometime in the 60s)...volume 1 (the chapter on sound) and compare it with this edition's treatment. I think you might like the older one a shade better.

Cheers
Vivek

 

[Doc Al]

read carefully!

Halliday, Resnick, and Walker, "Fundamentals of Physics", extended sixth edition. The erroneous definition of bulk modulus appears on page 286, in chapter 13.

It just so happens that I found an ancient edition (1966) sitting right here on the shelf. In the section that seems to correspond to what you are talking about, H&R say:

... so we now let p represent the change from the undisturbed pressure $p_0$. Then p replaces $\Delta p$, and ...

They aren't redefining the bulk modulus; they are just using "p" to represent the change from the undisturbed pressure $p_0$. Make sense?

 

[maverick280857]

Great Doc!