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The relation between T and g is given by
Answer
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Hint: In the question we are asked to find the relation between ‘T’ and ‘g’. Consider a mass “m” suspended on a wire of length ‘ ’ undergoes simple harmonic motion, ‘T’ is the time period, the time required to complete one oscillation and ‘g’ is the acceleration due to gravity (9.8m/s). To solve this we know the equation of time period; by squaring the known equation we can formulate the relation between ‘T’ and ‘g’.
Formula used:
Time period of a simple oscillation is given by
Complete answer:
To find the relation between ‘T’ and ‘g’,
Let us consider ‘ ’ to be the length of the pendulum.
As we know, time period is given by the equation
Squaring both sides of the equation, we get
From this equation we get,
Thus we can conclude that, , when l is unchanged.
So, the correct answer is “Option D”.
Additional Information:
Time of a simple pendulum derivation:
Consider a simple pendulum with a mass ‘m’ suspended on a wire of length ‘ ’.
For one oscillation the pendulum is displaced at an angle of ‘ ’ by ‘x’ distance.
Let be the time period at equilibrium.
When the pendulum oscillates, it is displaced at a small angle
For this small displacement , the restoring force acting will be
Restoring force=
Since the angle of displacement is very small here, we can approximate to
I.e.
Hence the force here can be rewritten as
Now let us consider the triangle ABC in the figure.
We know that sin of the angle is the ratio of the opposite side to the hypotenuse of the triangle. Since here , we can write this as
Here the opposite side of the angle is the displacement ‘x’ and the hypotenuse of the triangle is length ‘ ’ of the pendulum. Hence we can rewrite the equation as
Therefore the restoring force on the pendulum is
By Newton’s second law of motion, we have the equation of motion as
, Where ‘m’ is the mass of the body and ‘a’ is the acceleration.
We can rewrite this equation as
From the previous equation, we know that . Substituting this here, we get
Eliminating the common terms, we get
For a simple harmonic motion we know that,
On comparing both these equations, we get
By simplifying this,
Time period ‘T’ is given by the equation
Substitute the value of in this equation
Therefore time period,
Note:
This question can be solved by another method.
We know that, for a simple pendulum its angular frequency is given by
Time period of an oscillation can also be written as
By substituting the value of angular frequency ( ) in the above equation, we get
Thus we get
Hence we get the same solution.
Formula used:
Time period of a simple oscillation is given by
Complete answer:
To find the relation between ‘T’ and ‘g’,
Let us consider ‘
As we know, time period is given by the equation
Squaring both sides of the equation, we get
From this equation we get,
Thus we can conclude that,
So, the correct answer is “Option D”.
Additional Information:
Time of a simple pendulum derivation:
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Consider a simple pendulum with a mass ‘m’ suspended on a wire of length ‘
For one oscillation the pendulum is displaced at an angle of ‘
Let
When the pendulum oscillates, it is displaced at a small angle
For this small displacement
Restoring force=
Since the angle of displacement
I.e.
Hence the force here can be rewritten as
Now let us consider the triangle ABC in the figure.
We know that sin of the angle
Here the opposite side of the angle is the displacement ‘x’ and the hypotenuse of the triangle is length ‘
Therefore the restoring force on the pendulum is
By Newton’s second law of motion, we have the equation of motion as
We can rewrite this equation as
From the previous equation, we know that
Eliminating the common terms, we get
For a simple harmonic motion we know that,
On comparing both these equations, we get
By simplifying this,
Time period ‘T’ is given by the equation
Substitute the value of
Therefore time period,
Note:
This question can be solved by another method.
We know that, for a simple pendulum its angular frequency
Time period of an oscillation can also be written as
By substituting the value of angular frequency (
Thus we get
Hence we get the same solution.
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