Question

A guitar is being used in a church service by the musicians. When the guitar is plucked, it was observed that the velocity of the plucked strings is proportional to the powers of the length, mass, and tension of the strings according to the relation V=kL^aM^bT^c where k is constant and the values of a,b,c are not known. Calculate the values of a,b, and c.

Answer 1

The values of the exponents are:

a = 0

b = -1

c = 1

Therefore, the relation becomes: V = k * M^(-1) * T.

Step-by-step explanation:

To determine the values of the exponents a, b, and c in the relation V = kL^aM^bT^c, we can use dimensional analysis. In this approach, we analyze the dimensions of the variables involved (length, mass, and tension) and equate them to the dimensions of velocity (L/T).

Let's break down the dimensions of each variable:

V (velocity): [L]/[T]

L (length): [L]

M (mass): [M]

T (tension): [M][L]/[T]^2

Substituting these dimensions into the given relation:

[V] = [kL^aM^bT^c] = [k][L^a][M^b][T^c]

[L]/[T] = [k][L^a][M^b][M][L]/[T]^2

Now let's equate the dimensions on both sides of the equation:

[L]/[T] = [k][L^(a+1)][M^(b+1)][T^(c-2)]

Equating the dimensions of length on both sides:

1 = a + 1

Equating the dimensions of mass on both sides:

0 = b + 1

Equating the dimensions of time on both sides:

-1 = c - 2

c = 1

From the equation for mass, we found that b = -1, and from the equation for time, we found that c = 1. Using the equation for length, we have a = 0.

So, the values of the exponents are:

a = 0

b = -1

c = 1

Therefore, the relation becomes: V = k * M^(-1) * T.