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The time period of a simple pendulum depends on its effective length l and the local acceleration due to gravity g. What is the number of dimensionless parameters involved?
- a)Two
- b)One
- c)Three
- d)Zero
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The time period of a simple pendulum depends on its effective length ...
Number of variables (m) = 3 (Time period, length and gravity) Number of fundamental dimensions (n) = 3 (M, L and T) Number of dimensionless terms = m − n = 0
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The time period of a simple pendulum depends on its effective length ...
Understanding the Time Period of a Simple Pendulum
The time period of a simple pendulum is influenced by its effective length (l) and the local acceleration due to gravity (g). To determine the number of dimensionless parameters involved, we can analyze the relationship between these parameters.
Key Parameters
- Effective Length (l): This is the length from the pivot point to the center of mass of the pendulum.
- Acceleration due to Gravity (g): This is the force that pulls the pendulum downward, affecting its swinging motion.
Dimensional Analysis
To find dimensionless parameters, we look for combinations of l and g that yield a dimensionless quantity. The formula for the time period (T) of a simple pendulum is given by:
T = 2π√(l/g)
Here, T has dimensions of time, l has dimensions of length, and g has dimensions of acceleration (length/time^2).
When performing dimensional analysis, we can express T in terms of l and g, but since T is not dimensionless, we need to consider the relationships:
- The ratio of (l/g) is dimensionless because it combines length and acceleration, leading to a quantity that is independent of any specific units.
However, the time period itself is not dimensionless; it has its own dimension.
Conclusion
As a result, despite having parameters l and g, the only relevant dimensionless quantity that could arise is the ratio of l to g; however, this is not the focus here. Therefore, in the context of the time period's dependence on l and g, the conclusion is:
- There are zero dimensionless parameters involved in defining the time period of a simple pendulum, as T itself is a dimensional quantity.
Thus, the correct answer is option 'D'.
The time period of a simple pendulum is influenced by its effective length (l) and the local acceleration due to gravity (g). To determine the number of dimensionless parameters involved, we can analyze the relationship between these parameters.
Key Parameters
- Effective Length (l): This is the length from the pivot point to the center of mass of the pendulum.
- Acceleration due to Gravity (g): This is the force that pulls the pendulum downward, affecting its swinging motion.
Dimensional Analysis
To find dimensionless parameters, we look for combinations of l and g that yield a dimensionless quantity. The formula for the time period (T) of a simple pendulum is given by:
T = 2π√(l/g)
Here, T has dimensions of time, l has dimensions of length, and g has dimensions of acceleration (length/time^2).
When performing dimensional analysis, we can express T in terms of l and g, but since T is not dimensionless, we need to consider the relationships:
- The ratio of (l/g) is dimensionless because it combines length and acceleration, leading to a quantity that is independent of any specific units.
However, the time period itself is not dimensionless; it has its own dimension.
Conclusion
As a result, despite having parameters l and g, the only relevant dimensionless quantity that could arise is the ratio of l to g; however, this is not the focus here. Therefore, in the context of the time period's dependence on l and g, the conclusion is:
- There are zero dimensionless parameters involved in defining the time period of a simple pendulum, as T itself is a dimensional quantity.
Thus, the correct answer is option 'D'.
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The time period of a simple pendulum depends on its effective length l and the local acceleration due to gravity g. What is the number of dimensionless parameters involved?a) Twob) Onec) Threed) ZeroCorrect answer is option 'D'. Can you explain this answer?
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