(a)
Determine a differential equation for the oscillation
Answer to Problem 7.4CP
Explanation of Solution
Given information:
Cup diameter is
Density of the air is equal to
To find the differential equation, we should apply Newton’s second law of motion to the direction that the cup is moving,
The Newton’s second lay of motion is defined as,
The drag force is defined as,
Where,
Calculation:
Find the effective length of the pendulum,
Apply Newton’s second law of motion in the direction of the motion of the cup,
The sum of tangential forces will be,
Substitute for
Where,
Rearrange,
Conclusion:
The differential equation for the oscillation
(b)
Non-dimensionalize the obtained equation in sub-part a?
Answer to Problem 7.4CP
Explanation of Solution
Given information:
Cup diameter is
Density of the air is equal to
According to the sub-part a,
The oscillation
Calculation:
In below equation,
Assume,
Therefore,
In above equation
Therefore, take
Therefore, the dimensionless equation will be,
For small angle
Conclusion:
The dimensionless equation is equal to,
(c)
Determine the natural frequency for
Answer to Problem 7.4CP
Explanation of Solution
Given information:
Cup diameter is
Density of the air is equal to
According to the sub-part a,
The oscillation
Calculation:
In below equation,
Assume,
Therefore,
Neglect
For small angle
Rearrange,
Therefore,
Conclusion:
The natural frequency for small angles is equal to
(d)
Find the time required?
Answer to Problem 7.4CP
Pendulum is very lightly damped; therefore it takes almost
Therefore, the number of cycles to get down to
Explanation of Solution
Given information:
According to sub-part b,
Assume, the air at
According to the definitions,
Take the drag co-efficient as
Take the drag co-efficient as
Calculate the effective length,
Calculate the value of
Calculate the value of
Calculate the natural frequency,
To find the required time integrate numerically until,
Pendulum is very lightly damped; therefore it takes almost
Therefore, the number of cycles to get down to
Conclusion:
Pendulum is very lightly damped; therefore it takes almost
Therefore, the number of cycles to get down to
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Chapter 7 Solutions
Fluid Mechanics, 8 Ed
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