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1. Homework Statement
# A simple pendulum having charge q, mass m and effective length l is suspended from a rigid support between the plates of a charged capacitor whose plates are kept vertical. What is the time period of oscillation of the pendulum?
2. Homework Equations
Time period = 2(pi)[l/g’]^(1/2)
3. The Attempt at a Solution
I solved it in the following way:
Let x be the angular displacement at any instant. Let T be the tension in the string.
Restoring force(F) = (mgsinx + qEcosx)
For small values of x,
F= (mgx + qE)
But this won’t be a Simple harmonic motion. I am stuck here.
The solution as given in my book is as follows:
Tcosx = mg (1)
Tsinx = qE (2)
From (1) & (2),
T = [(mg)^2 + (qE)^2]^(1/2)
Effective g’= T/m
= [g^2 + (qE/m)^2]^(1/2)
Time period = 2(pi)[l/g’]^(1/2)
= 2(pi)[l/(g^2 + (qE/m)^2)^(1/2)]^(1/2)
As per them, the direction of effective g’ is inclined to the vertical by angle x. But effective g’ should be vertical, isn’t it? Does tension contribute to the restoring force experienced by the bob?I think it is the component of weight that contributes to the restoring force? But they haven’t used that concept at all to find the time period. Please help!
# A simple pendulum having charge q, mass m and effective length l is suspended from a rigid support between the plates of a charged capacitor whose plates are kept vertical. What is the time period of oscillation of the pendulum?
2. Homework Equations
Time period = 2(pi)[l/g’]^(1/2)
3. The Attempt at a Solution
I solved it in the following way:
Let x be the angular displacement at any instant. Let T be the tension in the string.
Restoring force(F) = (mgsinx + qEcosx)
For small values of x,
F= (mgx + qE)
But this won’t be a Simple harmonic motion. I am stuck here.
The solution as given in my book is as follows:
Tcosx = mg (1)
Tsinx = qE (2)
From (1) & (2),
T = [(mg)^2 + (qE)^2]^(1/2)
Effective g’= T/m
= [g^2 + (qE/m)^2]^(1/2)
Time period = 2(pi)[l/g’]^(1/2)
= 2(pi)[l/(g^2 + (qE/m)^2)^(1/2)]^(1/2)
As per them, the direction of effective g’ is inclined to the vertical by angle x. But effective g’ should be vertical, isn’t it? Does tension contribute to the restoring force experienced by the bob?I think it is the component of weight that contributes to the restoring force? But they haven’t used that concept at all to find the time period. Please help!
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