Question

The energy E of an oscillating body is simple harmonic motion depends on its mass m, frequency n and emplitude a using the method of dimensional analysis find the relation between E ,m, n and a .

Answer 1

Given:

The energy E of an oscillating body is simple harmonic motion depends on its mass m, frequency n and amplitude a

Therefore ,  [E] is proportional to $[m]^{p}$$[n]^{q}[a]^{r}$

Therefore, [E] = k $[m]^{p}$$[n]^{q}[a]^{r}$

To Find:

The relation between E ,m, n and a

Solution:

To find dimensions of :

(i) Mass :

• [m] =  [M}

(ii) Frequency :  n = 1/time

• [n] =  [$T^{-1}$]

(iii) Amplitude :

• [a] = [L]

(ii) Energy

• Energy = Force/time =  [Force]x[distance] = [Mass]x[Acceleration]x[Distance]
• [E] = [M][L][$T^{-2}$][L]
• [E] = [M][L²][$T^{-2}$]
• [E] = [m][a²][n²]

Therefore the relation between E , m , n , a is

• E = k m a² n² , where k is some constant, with no dimension.