In the figure, string 1 and string 2 have the same length L from the wall to the

Question

In the figure, string 1 and string 2 have the same length L from the wall to the wheel, however, in that length, string 1 has mass mi and string 2 has mass m?. Both strings are under tension owing to the hanging block of mass M [figure (a)]. Remember to write your answers in terms of the input parameters L, m_1, m_2, and M. What is the frequency of the fundamental harmonic produced by standing waves on string 1? The block is now divided into two blocks (with M_1 + M_2 = M) and the apparatus rearranged as shown in Figure (b). Find the values of M_1 and M_2 such that the two strings produce the same sound. What is the frequency of the sound produced with the arrangement described in part (c)? What is the average energy transported by the standing waves generated in each string?

Explanation / Answer

a)

Let the tension in the string be T

Then From the diagram we know that 2T =mg

Then the tension T =mg/2

For the string 1 and string 2 the freqency is given by

f =v/2L and we know that v =Sqrt(T/u) since u =m/L

Then the frequency of the first srting is given by f1 =(1/2L1)*Sqrt(T/m1/L1)

Similalry for the second string the freqnecy is given by f2 =(1/2L2)*Sqrt(T/m2/L2)

b)

Given that m1+m2 =M

They produce the same sound then m1L1 =m2L2

Then m1+m1L1/L2 =M

m1 =M*L2/(L1+L2)

and m2 =M*L1/(L1+L2)

d)

The average energy is given by

=(1/2)u*L*w2*v

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