Question

Time period \((T)\) of pendulum is directly proportional to the square root of length of string by which bob is attached to a fixed point and inversely proportional to the square root of gravitational constant ' \(g\) '. Time period of a bob is 3 seconds when the gravitational constant \(g\) is \(4 \mathrm{~m} / \mathrm{sec}^{2}\) and length of string is 9 metre, what is the time period of a bob having a string of length 64 metre and gravitational constant \(16 \mathrm{~m} / \mathrm{sec}^{2} ?\) (a) 4 seconds (b) 12 seconds (c) 16 seconds (d) 10 seconds

Short answer

Answer: (a) 4 seconds

Step-by-step solution

Write the proportionality relationship from the exercise statement.

According to the given exercise, the time period (T) of a pendulum is directly proportional to the square root of the length of the string (L) and inversely proportional to the square root of the gravitational constant (g). We can write this relationship as: \(T = k\frac{\sqrt{L}}{\sqrt{g}}\) Where k is the constant of proportionality.

Determine the value of the constant of proportionality (k) using the given values of T, L, and g for the first pendulum.

We are given that for the first pendulum, T = 3 seconds L = 9 meters g = 4 m/s² We can solve for k using these values: \(3 = k\frac{\sqrt{9}}{\sqrt{4}}\) Simplify the terms: \(3 = k\frac{3}{2}\) Solve for k: \(k = 2\)

Use the value of k to find the time period of the second pendulum with the new values of L and g.

Now that we know the constant of proportionality, we can use it to find the time period of the new pendulum with: L = 64 meters g = 16 m/s² Plug in these values and the value for k into the proportionality equation: \(T = 2\frac{\sqrt{64}}{\sqrt{16}}\) Simplify the terms and solve for T: \(T = 2\frac{8}{4}\) \(T = 4\) seconds So, the time period of the new pendulum is 4 seconds. Therefore, the correct answer is (a) 4 seconds.

Key concepts

Understanding Proportionality Relationships

When we talk about proportionality relationships in physics, we are referring to how certain physical quantities are related to each other. Proportionality indicates that when one quantity changes, the other changes in a consistent manner, either increasing or decreasing in response.

In the context of a pendulum, the time period (T), which is the time it takes for the pendulum to complete one full swing, exhibits a specific proportionality relationship with both the length of the string (L) and the gravitational constant (g). This relationship says that the time period of the pendulum is directly proportional to the square root of the length of the string. This means that if we were to increase the length of the pendulum's string, the time period would also increase. Conversely, it's inversely proportional to the square root of the gravitational constant, which implies that an increase in the gravitational constant would lead to a decrease in the time period.

Importance of Proportionality in Solving Physics Problems

Understanding these relationships is crucial for solving problems in physics because it allows us to predict how changing one variable will affect another. In the exercise, by establishing the proportionality between T, L, and g, we are able to calculate the new time period when either the string length or the gravitational constant is altered.

The Role of Gravitational Constant

The gravitational constant (often denoted as 'g'), is a key factor in the study of pendulum motion. It's the acceleration due to Earth's gravity, which pulls the pendulum towards the Earth's center. On the surface of the Earth, this value is approximately 9.81 m/s² but can vary slightly depending on altitude and latitude.

In our pendulum example, the gravitational constant is an inverse square root relationship with the time period. High school and introductory college physics courses often highlight that this value simplifies many gravitational force problems. It is also widely used in calculations involving the motion of objects under the influence of Earth's gravity.

Gravitational Constant's Effect on Time Period

If g is increased, the pendulum would accelerate more quickly and thus take less time to complete a swing, reducing its time period. This principle holds true even beyond the classroom, assisting engineers and scientists in fields such designing pendulums for clocks or measuring gravitational variations in different parts of the world.

Square Root Relationships Made Simple

A square root relationship in mathematics indicates that a quantity is related to the square root of another quantity. This kind of relationship is not linear—it does not follow a straight line when graphed. Instead, it forms a curve that shows a gradual change rather than a constant rate of change.

In the case of the pendulum, both the length of the string and the gravitational constant are in a square root relationship with the time period. This means we don't expect the time period to double when we double the length or gravitational constant due to the nature of square roots.

Applying Square Root Relationships

For instance, if the length of the string is quadrupled, since we're interested in its square root, the actual effect on the pendulum's time period is that it will double, not quadruple. A solid grasp of square root relationships is essential for students hoping to excel in physics, as it enables the smooth translation of theoretical formulas into practical solutions, as demonstrated in the exercise under consideration.