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A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period?
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A uniform circular disc of radius R oscillates in a vertical plane abo...
Uniform Circular Disc Oscillation
In a uniform circular disc oscillation, the distance of the axis of rotation from the centre plays a crucial role in determining the period of the oscillation. Let us find out the distance at which the period is minimum and the corresponding value of the period.
Period of Oscillation
The period of oscillation of a uniform circular disc of radius R is given by:
T = 2π√(I/mgR)
where I is the moment of inertia of the disc, m is its mass, g is the acceleration due to gravity, and R is the radius of the disc.
Determining the Distance of the Axis of Rotation
Let us assume that the distance of the axis of rotation from the centre of the disc is x. Then, the moment of inertia of the disc about the axis of rotation is given by:
I = (1/2)mR^2 + (1/4)m(x^2 + R^2)
Substituting this value of I in the expression for T, we get:
T = 2π√[(1/2)R^2 + (1/4)(x^2 + R^2)]/(gR)
To find the value of x for which T is minimum, we differentiate T with respect to x and equate it to zero:
dT/dx = πx/2√[(1/2)R^2 + (1/4)(x^2 + R^2)]^3 = 0
Solving this equation, we get:
x = R/√2
Value of the Period
Substituting x = R/√2 in the expression for T, we get:
T = 2π√(3/4)R/g
This is the minimum value of the period of oscillation of the uniform circular disc. Its numerical value can be calculated using the given values of R and g.
Conclusion
Thus, we have determined the distance of the axis of rotation from the centre of the uniform circular disc for which the period of oscillation is minimum, and also found the corresponding value of the period.
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A uniform circular disc of radius R oscillates in a vertical plane abo...
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A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period?
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A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period?.
A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A uniform circular disc of radius R oscillates in a vertical plane about a horizontal Axis. find the distance of the axis of rotation from the centre for which the period is minimum. what is the value of this period?.
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