Concept explainers
(a)
The period of small oscillations.
(a)
Explanation of Solution
Given:
The distance between center of sphere and point of support is
The period of simple pendulum is
Introduction:
Time period is the time taken to cover one oscillation. The body resists any
The moment of inertia of sphere is
Write the expression for moment of inertia using parallel axis theorem.
Here,
Substitute
Write the expression for object in term of period of pendulum.
Here,
Substitute
Here,
Conclusion:
Thus, the period of small oscillations is
(b)
The period of small oscillations when radius is much smaller than length of pendulum.
(b)
Explanation of Solution
Given:
The distance between center of sphere and point of support is
The period of simple pendulum is
Introduction:
Time period is the time taken to cover one oscillation. The body resists any angular acceleration due to inertia present it which is known as moment of inertia.
The moment of inertia of sphere is
Write the expression for moment of inertia using parallel axis theorem.
Here,
Substitute
Write the expression for object in term of period of pendulum.
Here,
Substitute
Expand the term
Substitute
Here,
Conclusion:
Thus, the period of small oscillations when radius is much smaller than length of pendulumis
(c)
The error in the time period of oscillations; the value of radius for
(c)
Explanation of Solution
Given:
The distance between center of sphere and point of support is
The radius of bob in first case is
The error for second case is
Introduction:
Time period is the time taken to cover one oscillation. The body resists any angular acceleration due to inertia present it which is known as moment of inertia.
The moment of inertia of sphere is
Write the expression for moment of inertia using parallel axis theorem.
Here,
Substitute
Write the expression for object in term of period of pendulum.
Here,
Substitute
Expand the term
Substitute
Here,
Calculate the error of time period for approximation of
Substitute
Conclusion:
For case1:
Substitute
For case2:
Substitute
Thus, the error in the time period of oscillation is
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Chapter 14 Solutions
Physics for Scientists and Engineers
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