Question

1. To dering the equations for period (T) of a simple Pendulum assuming that it may depend on mass of the bob (m) length of the pendulum. (L) and acceleration due to gravity (g) of the given place. ​

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Derive an expression for time ...

Derive an expression for time period (t) of a simple pendulem, which may depend upon : mass of bob (m), length of pendulum (I) and acceleration due to gravity(g).

Updated On: 1-6-2020



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408.0 K



1x1.5x2x



Answer

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Solution

Let t∝malbgct∝malbgc where a, b, c are the dimensions. Or t=kmalbgc…(4)t=kmalbgc…(4)

where k is dimensionless constant of proportionality.

Writing the dimensions in terms of M, L, T on either side of (4), we get

[M0L0T1]=MaLb(LT−2)c=MaLb+cT−2c[M0L0T1]=MaLb(LT-2)c=MaLb+cT-2c

Applying the principle of homogeneity of dimensions, we get

a=0b+c=0....(5)a=0b+c=0....(5)

−2c=1orc=−12-2c=1orc=-12

From (5),b=−c=−(−12)=12b=-c=-(-12)=12

Putting the value of a,b,c, inPutting the value of a,b,c, in (4), we getwe get

t=km0I1/2g−1/2t=km0I1/2g-1/2