The time period of a pendulum depends on mass , length and acceleration due to gravity. Derive the relation among those physical quantitieS
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Answer:
Let Time period =T
Mass of the bob = m
Acceleration due to gravity = g
Length of string = L
Let T \alpha m ^{a}g ^{b}L ^{c}Tαm
a
g
b
L
c
[T] \alpha [m] ^{a}[g] ^{b}[L] ^{c}[T]α[m]
a
[g]
b
[L]
c
M^{0}L^{0}T^{1}=M^{a}L^{b}T^{-2b}L^{c}M
0
L
0
T
1
=M
a
L
b
T
−2b
L
c
M^{0}L^{0}T^{1}=M^{a}L^{b+c}T^{-2b}M
0
L
0
T
1
=M
a
L
b+c
T
−2b
⇒a=0 ⇒ Time period of oscillation is independent of mass of the bob
-2b=1
⇒b=-\frac{1}{2}
2
1
b+c = 0
-\frac{1}{2}
2
1
+ c =0
c=\frac{1}{2}
2
1
Giving values to a,b and c in first equation
T \alpha m ^{0}g ^{- \frac{1}{2} }L ^{ \frac{1}{2} }Tαm
0
g
−
2
1
L
2
1
T \alpha \sqrt{ \frac{L}{g} }Tα
g
L
The real expression for Time period is
T =2 \pi \sqrt{ \frac{L}{g} }T=2π
g
L
Therefore time period of oscillation depends only on gravity and length of the string.
Not on mass of the bob.