The bob in a simple pendulum of length l is released at t = 0 from the of small angular displacement theta_(0). Linear displacement of the bob at any time t from the mean position is given by :
Answers
Answer:
x= lcoswt
=
l
L
g
tcosθ
Explanation:
The bob in a simple pendulum of length ℓis released at
t=
0
from the position of small angular displacement
q
. Linear displacement of the bob at any time tfrom the mean position is given by
A .
ℓθcos
l
g
t
B .
ℓ
l
g
tcosθ
C .ℓgsinθ
D .
ℓθsin
l
g
t
December 26, 2019
A
Dear student,
Let us consider a simple pendulum of length l. Let is be shifted to a rest position at some angle ∅ with the vertical. Let the that position be the initial time stamp.
Now consider the mean position that is the vertical position of bob right in the bottom-mid of the pendulum system. In this position, let the it be attached to the main system with a gap of distance H.
Also, let the displacement of the bob at rest with angle ∅ be h. Now, in order to find the linear displacement when the bob is at the mean position, we consider the velocity at the mean position.
By work-energy theorem or by conservation of mechanical energy theorem, we know that velocity at mean position is given by;
v = √2gl (1 - cos∅)
We consider the formula to have an idea as of what the linear displacement is expected to come like. So, it will come up with some appreciable value with cos component attached to it.
In the attached image, the red line indicated the linear displacement of the bob. So, how do we obtain its value? See, in order to find its value we have subtract it from the other given values as;
Since, the length of the string connected to bob remains constant, hence its length remains l. So, vertical component of the l in the upper triangular region comes out to be l cos∅.
Hence, length of red region ( required region ) is given by;
Red region = l - l cos∅
Red region = l ( 1 - cos∅ )
So, linear displacement of the bob at any time t from the mean position is given by l ( 1 - cos∅ ).
