Assertion: The angular velocity of the moon revolving around the earth is more than that of the earth revolving around the Sun.
Reason: The time taken by the moon to revolve around the earth is less than the time taken by the earth to revolve around the sun.
Reason: The time taken by the moon to revolve around the earth is less than the time taken by the earth to revolve around the sun.
- Both Assertion (A) and Reason (R) are the true and Reason (R) is a correct explanation of Assertion (A).
- Both Assertion (A) and Reason (R) are the true but Reason (R) is not a correct explanation of Assertion (A).
- Assertion (A) is true and Reason (R) is false.
- Assertion (A) is false and Reason (R)
The Correct Option is A
Approach Solution - 1
The question involves understanding the concept of angular velocity within the context of celestial bodies—specifically, the Moon revolving around the Earth and the Earth revolving around the Sun.
Angular Velocity Explanation:
- Angular velocity (\(\omega\)) is defined as the rate of change of the angular displacement and is usually measured in radians per second.
- The formula to calculate angular velocity is \(\omega = \frac{2\pi}{T}\), where \(T\) is the period of revolution.
Comparison:
- For the Moon revolving around the Earth, the time period \(T_{\text{moon}}\) is approximately 27.3 days.
- For the Earth revolving around the Sun, the time period \(T_{\text{earth}}\) is approximately 365.25 days.
Since \(T_{\text{moon}} < T_{\text{earth}}\), it follows that the Moon has a greater angular velocity because:
\(\omega_{\text{moon}} = \frac{2\pi}{T_{\text{moon}}} > \frac{2\pi}{T_{\text{earth}}} = \omega_{\text{earth}}\)
Analysis of Assertion and Reason:
- Assertion (A): The angular velocity of the moon revolving around the Earth is more than that of the Earth revolving around the Sun. This is true, as calculated above.
- Reason (R): The time taken by the Moon to revolve around the Earth is less than the time taken by the Earth to revolve around the Sun. This is true and it properly explains why the angular velocity of the Moon is greater.
Thus, the correct answer is: Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
Approach Solution -2
The angular speed \( \omega \) of an object is given by:
\(\omega = \frac{2\pi}{T},\)
where \( T \) is the time period.
For the moon:
- \( T_{\text{moon}} = 27 \, \text{days} \).
For the earth:
- \( T_{\text{earth}} = 365 \, \text{days} \).
Since the moon takes less time to complete one orbit around the earth compared to the earth's revolution around the sun, \( T_{\text{moon}} < T_{\text{earth}} \). Therefore:
\(\omega_{\text{moon}} > \omega_{\text{earth}}.\)
This makes both the assertion and the reason correct, and the reason is the correct explanation of the assertion.
The correct option is (A) : Both Assertion (A) and Reason (R) are the true and Reason (R) is a correct explanation of Assertion (A).
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Concepts Used:
Gravitational Potential Energy
The work which a body needs to do, against the force of gravity, in order to bring that body into a particular space is called Gravitational potential energy. The stored is the result of the gravitational attraction of the Earth for the object. The GPE of the massive ball of a demolition machine depends on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between GPE and the mass of an object. More massive objects have greater GPE. Also, there is a direct relation between GPE and the height of an object. The higher that an object is elevated, the greater the GPE. The relationship is expressed in the following manner:
PEgrav = mass x g x height
PEgrav = m x g x h
Where,
m is the mass of the object,
h is the height of the object
g is the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.