Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). 
Assertion (A): Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. 
Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements. 
choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are true but (R) is not the correct explanation of (A)
• B. (A) is false but (R) is true
• C. Both (A) and (R) are true and (R) is the correct explanation of (A)
• D. (A) is true but (R) is false

Hint:

The time period of a simple pendulum is inversely related to the square root of the acceleration due to gravity.

Answer 1

The Correct Option is A

To solve the problem, let's analyze the given statements:

• Assertion (A): The time period of a simple pendulum is longer at the top of a mountain than at the base of the mountain.
• Reason (R): The time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

The formula for the time period (T) of a simple pendulum is:
T = 2π√(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity. From this formula, we see that the time period is inversely related to the square root of g. As the altitude increases, like at the top of a mountain, the value of g decreases slightly because of the increase in distance from the center of the Earth. This leads to a longer time period (T) at the mountain top compared to the base. Thus, Assertion (A) is true.
The Reason (R) correctly states that the time period decreases with an increase in g, which is mathematically accurate. However, while this reason is true, it does not specifically explain why the time period is longer at a mountain top. This is due to the decrease in g at higher altitudes, which is only indirectly related to the reason given.
Therefore, the correct answer is: Both (A) and (R) are true but (R) is not the correct explanation of (A).

Answer 2

Step 1: Understand the given statements.
Assertion (A): The time period of a simple pendulum is longer at the top of a mountain than at the base of the mountain.
Reason (R): The time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

Step 2: Recall the formula for the time period of a simple pendulum.
The time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{l}{g}} \] where \( l \) is the length of the pendulum and \( g \) is the acceleration due to gravity.

Step 3: Analyze the effect of gravity at the top of a mountain.
The value of acceleration due to gravity \( g \) decreases with height from the Earth's surface. Therefore, at the top of a mountain, \( g \) is smaller compared to its value at the base of the mountain.
From the formula above, when \( g \) decreases, the time period \( T \) increases. Hence, the pendulum takes a longer time to complete one oscillation at the mountain top.

Step 4: Evaluate the truth of Assertion (A) and Reason (R).
- Assertion (A) is true because the time period is indeed longer at the top of the mountain due to the smaller value of \( g \).
- Reason (R) is also true because the time period \( T \) is inversely proportional to the square root of \( g \); an increase in \( g \) decreases \( T \), and vice versa.

Step 5: Determine if (R) correctly explains (A).
Although both statements are true, (R) is a general statement about the relationship between \( T \) and \( g \). It does not directly explain the specific reason why \( g \) is smaller at the mountain top (which is due to increased distance from the Earth's center). Therefore, (R) does not correctly explain (A).

Final Answer:
Both (A) and (R) are true but (R) is not the correct explanation of (A).