Question

write the equation of motion of a simple pendulum and use it to obtain the condition under which its motion is simple harmonic in nature obtain an expression for its time period under this condition
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Answer 1

Answer:

Explanation:

Ideal simple pendulum is defined as a heavy point mass suspended from a rigid support by a weightless and inextensible string and set oscillating under gravity through a small angle in a vertical plane.

Let a simple pendulum of length L suspended from a rigid support O. Let it is displaced by a small angle θ in a vertical plane and released. Resolving mg into horizontal and vertical components at point B as mgcosθ and mgsinθ respectively.

We see that restoring force

F=−mgsinθ

​  

If  'θ ' is small then

sinθ=θ=  

L

x

​  

 

F=−mgθ

   =−mg  

L

xWe see that F∝(−x)

Since F is directly proportional to negative of displacement so motion of a simple pendulum is in linear S.H.M.

So        acceleration =  

m

F

​ =  

L

−acceleration per unit displacement

∣

∣

∣

∣

​  

 

x

a

​  

 

∣

∣

∣

∣

​  

=  

L

g

​T=  

accelerationperunitdisplacement

​  

 

2π

​  

 

   =  

L

g

​  

 

​  

 

2π

​  

 

T=2π  

g

L

​  

 

 

 

 

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