Question

Suppose that the oscillatións of a simple pendulum depend on (5) (i) mass of the bob (m), (ii) length of the string (l), (iii) acceleration due to gravity (g) Dimensionally show which of the above factors have influence upon the period and in what way?​

Answer 1

$T = K {l}^{x} m {}^{y} g {}^{z}$

Where K is dimensional Constant.

x, y, z are the exponents of l , m and g respectively.

Dimensional analysis :

$(M {}^{0} L {}^{0} T {}^{1} ) = (M {}^{y} L {}^{x + z} T {}^{ - 2z} )$

Equating :-

y= 0

x + z = 0 _____(1)

-2z = 1

$z = - \frac{1}{2}$

put the value of Z in equation __(1)

$x = \frac{1}{2}$

Therefore,

$T = K \sqrt{ \frac{l}{g} }$

where k is 2π

$T = 2 \pi \sqrt{ \frac{l}{g} }$

●Hope it helps you

Answer 2

Explanation:

• Show that   cannot end with the digits   or 
• for any natural number