Question

The graph of
\[ \ln\left(\frac{R}{R_0}\right)\ \text{versus}\ \ln A \] is
where \(R\) is the radius of a nucleus, \(A\) is its mass number, and \(R_0\) is constant.

• A. A straight line
• B. A circle of radius \(R\)
• C. A parabola
• D. An ellipse

Hint:

The nuclear radius relation \(R=R_0A^{1/3}\) becomes linear when logarithms are taken on both sides.

Answer 1

The Correct Option is A

Step 1: Write the relation between nuclear radius and mass number.
The radius of a nucleus is given by
\[ R=R_0A^{1/3} \]
where
\[ R_0=\text{constant} \] and \(A\) is the mass number.

Step 2: Rearrange the equation.
Divide both sides by \(R_0\):
\[ \frac{R}{R_0}=A^{1/3} \]
Now take natural logarithm on both sides:
\[ \ln\left(\frac{R}{R_0}\right)=\ln\left(A^{1/3}\right) \]
Using logarithmic property,
\[ \ln\left(A^{1/3}\right)=\frac13\ln A \]
Thus,
\[ \ln\left(\frac{R}{R_0}\right)=\frac13\ln A \]

Step 3: Identify the graph.
The above equation is of the form
\[ y=mx \]
which represents a straight line passing through the origin with slope \(\frac13\).

Step 4: Final conclusion.
Hence, the graph is
\[ \boxed{\text{A straight line}} \]