Problem 17 - Master Resource Book in JEE Main Physics

Question

Assuming the earth to be a sphere of uniform mass density, how much would body weigh half way down to the centre of earth if it weighed \(250 \mathrm{~N}\) on the surface? (a) \(225 \mathrm{~N}\) (b) \(325 \mathrm{~N}\) (c) \(100 \mathrm{~N}\) (d) \(125 \mathrm{~N}\)

Step-by-Step Solution

Answer

The weight halfway is 125 N, option (d).

Step 1: Understanding the Problem

We are given the weight of a body on the Earth's surface as \(250 \mathrm{~N}\). We need to calculate its weight halfway to the center of the Earth. Earth's mass is uniformly distributed, so we can use Gauss's law for gravitation.

Step 2: Conceptualize Gravitational Force Inside Earth

Inside a sphere of uniform density, the gravitational force at a depth is proportional to the distance from the center. If \(R\) is the radius of the Earth, then halfway means \( \frac{R}{2} \). So, the gravitational force would be \( F_{new} = F_{surface} \left( \frac{r}{R} \right) \) where \( r = \frac{R}{2} \).

Step 3: Expressing the Relationship

Given the proportionality from step 2: \[ F_{new} = F_{surface} \left( \frac{\frac{R}{2}}{R} \right) = F_{surface} \left( \frac{1}{2} \right) \]Thus, the weight of the body halfway down is half of what it was on the surface.

Step 4: Calculate the New Weight

Substituting the given surface weight, \( F_{surface} = 250 \mathrm{~N} \), and using the relationship:\[ F_{new} = 250 \times \frac{1}{2} = 125 \mathrm{~N} \]

Step 5: Select the Correct Answer

The calculated weight halfway to the center of the Earth is \(125 \mathrm{~N}\). Therefore, the correct option is (d).

Key Concepts

Gauss's Law for Gravitation

We all know about the amazing gravitational force that pulls us towards the Earth. But did you know there is a special law that helps us understand gravity in even more detail? It's called Gauss's Law for Gravitation. This law lets us calculate gravitational force in various scenarios, especially when dealing with spherical objects. It’s particularly useful when we consider the gravitational effect enclosed by a symmetric shape like a sphere.

This law helps us determine how gravity changes, especially inside spheres with uniform mass density, like the Earth. When you're deep inside the Earth, you only need to consider the mass beneath you for calculating gravity. This makes it possible to assume simpler models when analyzing problems.

• Useful for large objects that are roughly spherical.
• Essential for understanding how gravitational forces act within these objects.
• Allows for simpler calculations when thinking about forces inside spherical bodies.

Remember, Gauss's Law for Gravitation gives us an elegant tool to simplify and solve gravitational problems, helping us to calculate weights at different depths inside the Earth.

Uniform Mass Density

Uniform mass density might sound complex, but it’s pretty straightforward. Imagine a sphere like the Earth having the same amount of mass everywhere inside it. That’s what we mean when we say "uniform mass density." This assumption is helpful because it allows us to predict gravitational forces inside the sphere more easily.

For Earth, although it's not perfectly uniform, the concept of uniform mass density is often used to simplify calculations. It lets us assume that every section of the Earth has the same amount of mass. So, if you dig down halfway to the Earth's center, you can figure out how gravity affects you by only considering the mass up to that point. This uniform spread helps in applying Gauss’s law effectively and makes mathematical calculations easier.

• Assumption simplifies mathematical models.
• Provides a consistent way to apply physical laws.
• Used widely to simplify problem-solving in gravitational physics.

By thinking about Earth as having uniform mass density, you can predict things like weight changes as you move towards or away from its center.

Weight Variation Inside Earth

Weight varies as you move beneath the Earth's surface. But why does this happen? Simply put, weight is the force exerted by gravity on an object. If you go beneath the Earth’s surface, your weight diminishes. This happens because the force of gravity is only based on the mass below your feet postitioned at any given point.

As you move toward the Earth’s center, the gravitational pull that affects you comes from the mass beneath you inside the Earth. This is where Gauss’s law for gravitation shines, helping us estimate gravitational forces at different depths effectively.

• Weight decreases as you go deeper because less mass is below you exerting gravity.
• Weight at Earth’s center in theory could be zero, with gravity pulling equally from all sides.
• Important for understanding how underground environments might affect weight-based measurements.

Understanding weight variation can be key in fields like geology and astrophysics where knowing the effect of gravity in different environments matters.