Problem 813 - JEE Physics

Question

If the radius of the earth were to shrink by \(1 \%\) its mass remaining the same, the acceleration due to gravity on the earth's surface would (A) decrease by \(2 \%\) (B) remain Unchanged (C) increase by \(2 \%\) (D) increases by \(1 \%\)

Step-by-Step Solution

Answer

If the radius of the Earth were to shrink by 1% while its mass remains the same, the acceleration due to gravity on the Earth's surface would increase by approximately 2.03%. The answer is closest to (C) increase by 2%.

Step 1: Relevant Formulas

In order to solve the problem, we will need the formula for gravitational acceleration, which is: \( g = \frac{GM}{r^2} \) Where: - \(g\) is the gravitational acceleration - \(G\) is the gravitational constant (approximately \(6.674 \times 10^{-11} m^3 kg^{-1} s^{-2}\)) - \(M\) is the mass of the body (Earth in our case) - \(r\) is the radius of the body

Step 2: Initial Gravitational Acceleration

First, let's determine the initial gravitational acceleration before the radius shrinks. We will define the initial values as: - Radius: \(r\) - Gravitational Acceleration: \(g\) Now we can express the initial gravitational acceleration as: \( g = \frac{GM}{r^2} \)

Step 3: Shrunk Radius and New Gravitational Acceleration

The exercise tells us that the radius of the Earth shrinks by 1%. Therefore, the new radius will be: \( r' = r - 0.01r = 0.99r \) Now let's find the new gravitational acceleration (\(g'\)) with the new radius: \(g' = \frac{GM}{(0.99r)^2} = \frac{GM}{0.9801r^2}\)

Step 4: Comparing New and Old Gravitational Acceleration

Now we will compare the new gravitational acceleration (\(g'\)) to the old one (\(g\)), by taking the ratio of \(g'\) to \(g\): \(\frac{g'}{g} = \frac{GM}{0.9801r^2} \times \frac{r^2}{GM} = \frac{1}{0.9801} \approx 1.0203\) Which indicates that the gravitational acceleration increased. To find the percentage increase, we can subtract 1 and multiply by 100: \((1.0203 - 1) \times 100 = 2.03 \% \)

Step 5: Conclusion

Therefore, if the radius of the Earth were to shrink by 1% while the mass remains the same, the acceleration due to gravity on the Earth's surface would increase by approximately 2.03%. The answer is closest to (C) increase by 2%.

Key Concepts

Gravity and Mass

The concept of gravity is beautifully intertwined with mass. Gravity can be thought of as the force that pulls objects toward each other. Massive objects like planets have a strong gravitational pull.

• The gravitational pull of the Earth is directly related to its mass.
• More massive objects exert a stronger gravitational force.

However, it's interesting to note that changes in mass directly affect gravity, but changes in gravity don't necessarily mean a change in mass.The force of gravity is calculated using the gravitational constant, the mass of the objects, and the distance between their centers. This relationship is crucial because it helps us understand how different factors, like mass and distance, influence gravitational forces. Gravitational force can be calculated using Newton's law of universal gravitation, which is represented by the equation: \[ F = \frac{GM_1M_2}{r^2} \]Where: - \( F \) is the force of gravity. - \( G \) is the gravitational constant. - \( M_1 \) and \( M_2 \) are the masses of the objects. - \( r \) is the distance between the centers of the two objects. This formula shows how essential mass is in determining the strength of gravitational forces.

Earth's Radius

The radius of the Earth plays a significant role in the gravitational acceleration we experience. The radius is essentially the distance from the Earth's center to its surface. Changes in this radius can affect gravitational acceleration.

• The current average radius of the Earth is about 6,371 kilometers.
• If the radius changes but the mass remains constant, the gravitational force will adjust accordingly.

For calculating gravitational acceleration, the radius is squared in the denominator of our central formula: \[ g = \frac{GM}{r^2} \]When the Earth's radius decreases, the denominator in our equation reduces, leading to an increase in the value of \( g \). Conversely, increasing the radius would decrease the gravitational acceleration. This calculation highlights how the shape and size of the Earth, especially its radius, are essential in the interplay of gravitational forces that affect every object on the planet's surface.

Percentage Change in Gravity

When we talk about changes in gravity, we're often interested in percentages. It's a convenient way to share how much something increases or decreases relative to its original value. In the context of Earth's gravitational acceleration, calculating the percentage change is a straightforward yet insightful exercise. 1. First, we need to know the initial and new values for gravitational acceleration. Consider the formula: \( g' = \frac{GM}{(0.99r)^2} \)2. Next, we compare this new value to the original gravitational acceleration \( g \) using the ratio:\[ \frac{g'}{g} \approx 1.0203 \] 3. Finally, the percentage change is simply the difference from 1, multiplied by 100 to convert it to a percentage. The calculation we performed showed that a 1% reduction in Earth's radius leads to roughly a 2.03% increase in gravitational acceleration. This demonstrates how sensitive the gravitational pull is to even small changes in the Earth's physical properties. Understanding these calculations empowers students to grasp how minor adjustments in astronomical metrics can have substantial impacts.