Question #b0f77

1 Answer
Jan 29, 2016

rho_(Cu)=7575"kg/m"^3

or 7.57"g/cm"^3

Explanation:

The key to this question is to find the volume of the copper block.

Once we get this we can find the density since

density = mass / volume and the mass is given.

The relationship between the frequency f of the note and the tension T in the wire is given by:

f=(1)/(2L)sqrt((T)/(mu))

L is the length of the wire

mu is the mass per unit length of the wire

We will assume that these remain constant.

Diagram (a)

MFDocsMFDocs

Since the wire is in tune with the fork we can say:

f_1=96"Hz"

If the block is immersed in water then Archimedes tells us that it will experience an upthrust which is equal to the weight of water displaced.
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This means that the tension in the wire will be slightly reduced which will result in the frequency being reduced.

(Think what happens to the note of a guitar string if it is loosened).

The 2 notes sounded together make "beats". The beat frequency is given by :

f_b=Deltaf

Where Deltaf is the magnitude of the difference between the 2 frequencies. Since we know that the new frequency must be lower we can say:

f_2=96-5.6=90.4"Hz"

So now we set up 2 equations to eliminate the constants:

f_1=(1)/(2L)sqrt((T_1)/(mu))" "color(red)((1))

and

f_2=(1)/(2L)sqrt((T_2)/(mu))" "color(red)((2))

Divide color(red)((1)) by color(red)((2))rArr

(f_1)/(f_2)=(cancel((1)/(2L))sqrt((T_1)/(cancel(mu))) )/(cancel((1)/(2L))sqrt((T_2)/(cancel(mu))))

This simplifies down to:

(f_1)/(f_2)=sqrt((T_1)/(T_2))

Putting in the numbers:

96/90.4=sqrt((10xx9.8)/(T_2))=1.0619

:. 1.0619 = sqrt(98/T_2

:. 1.0619^2=98/T_2=1.1277

:.T_2=98/1.1277=86.9"N"

Diagram (b)

MFDocsMFDocs

You can see from diagram (b) that from Archimedes the upthrust U provided by the water is given by:

U=98-86.9=11.1"N"

This equals the weight of water displaced so:

mg=11.1

:.m=11.1/9.8=1.132"kg"

We are given the density rho of the water so:

rho=m/v

:.v=m/rho=1.132/1000=0.00132"m"^3

This must equal the volume of the copper block.

:.rho_(Cu)=m/v=10/0.00132=7575.7"kg/m"^3