Consider two solid spheres of radii R1 = 1 m, R2 = 2 m and masses M1 and M2, respectively. The gravi

Q: Why does the graph start with a straight line?

Inside a uniform solid sphere, the gravitational field is directly proportional to the distance from the center ($E = \frac{GMr}{R^3}$). This linear relationship creates the straight line starting from the origin.

Q: Why does the field decrease after the peak?

Outside the sphere, it behaves like a point mass concentrated at the center. The field follows the inverse-square law ($E = \frac{GM}{r^2}$), which causes the curve to drop off.

Q: Can we find the exact mass of the spheres?

We can find the value of $GM$, but to find the exact mass $M$, we would need to divide by the universal gravitational constant $G$. Since we only need the ratio, $G$ cancels out!

Q: What if the spheres were hollow shells?

For a hollow spherical shell, the gravitational field inside is exactly zero. The graph would be flat at zero until $r=R$, and then jump to the maximum value before decreasing.

Q: Why did we use $E = \frac{GM}{R^2}$ instead of the internal field formula?

The peaks on the graph give us the maximum field, which occurs exactly at the surface ($r=R$). The surface formula is the simplest and most direct way to relate the given peak values to the mass.

Question

Consider two solid spheres of radii $R_1 = 1	ext{ m}$, $R_2 = 2	ext{ m}$ and masses $M_1$ and $M_2$, respectively. The gravitational field due to sphere 1 and 2 are shown. The value of $\frac{M_1}{M_2}$ is

Options:

(A) $\frac{1}{3}$

(B) $\frac{1}{2}$

(C) $\frac{1}{6}$

(D) $\frac{2}{3}$

Sigma Percentile · Accepted Answer

Reading graphs is a superpower in physics. When you look at a graph, you aren't just seeing lines and curves; you are looking at the hidden physical properties of an object. In this problem, we are given the gravitational field graphs of two solid spheres, and we need to find the ratio of their masses. Let's decode the visual data!

### The Anatomy of the Graph

Look closely at the shape of the curves. For a uniform solid sphere, the gravitational field behaves differently inside and outside the sphere.

Inside the sphere ($r < R$), the field increases linearly with distance from the center: $E = \frac{GMr}{R^3}$. This is why both curves start as straight lines from the origin.

Outside the sphere ($r > R$), the sphere behaves like a point mass, and the field drops off according to the inverse-square law: $E = \frac{GM}{r^2}$.

The most crucial point on this graph is the **peak**. The peak represents the exact boundary of the sphere—its surface ($r = R$). At this point, the gravitational field is at its absolute maximum:

$$E_{	ext{max}} = \frac{GM}{R^2}$$

### Extracting Data for Sphere 1

Let's focus on the first curve. The peak occurs at a radius $R_1 = 1	ext{ m}$, and the maximum field value there is $E_1 = 2$.

We can plug these values directly into our surface field formula:

$$2 = \frac{GM_1}{1^2}$$

Solving this gives us a clean expression for the mass parameter of the first sphere:

$$GM_1 = 2$$

### Extracting Data for Sphere 2

Now, let's do the exact same thing for the second curve. The peak is located at $R_2 = 2	ext{ m}$, and the maximum field is $E_2 = 3$.

Substituting these into the formula:

$$3 = \frac{GM_2}{2^2}$$

Since $2^2 = 4$, we multiply both sides by 4 to isolate $GM_2$:

$$GM_2 = 3 	imes 4 = 12$$

### The Final Ratio

We are asked to find the ratio of the masses, $\frac{M_1}{M_2}$. Notice that we don't actually need the value of the universal gravitational constant $G$, because it will beautifully cancel out when we take the ratio!

$$\frac{M_1}{M_2} = \frac{GM_1}{GM_2}$$

Plugging in the values we just found:

$$\frac{M_1}{M_2} = \frac{2}{12}$$

Simplifying the fraction, we get our final answer:

**$$\frac{M_1}{M_2} = \frac{1}{6}$$**

By simply understanding that the peak of the graph corresponds to the surface of the sphere, a seemingly complex graphical problem turns into a straightforward substitution exercise. Always look for the peaks!

Animated Solution for Physics - Gravitation: Consider two solid spheres of radii $R_{1} = 1 m$ , $R_{2} = 2 m$ and masses $M_{1}$ and $M_{2}$ , respectively. The gravitational field due to sphere 1 and 2 are shown. The value of $\frac{M _{1}}{M _{2}}$ is

Select Answer:

Visualized Solution

$E_{max}$ at Surface

Formula for $E_{max}$

Analyzing Sphere 1

Analyzing Sphere 2

Calculating the Ratio

The Sigma Insight: Gravitational Force

The Anatomy of the Graph

Extracting Data for Sphere 1

Extracting Data for Sphere 2

The Final Ratio

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