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: Recall the nuclear radius law.
The radius of a nucleus depends on its mass number $A$ as \[ R = R_0 A^{1/3}, \] where $R_0$ is a constant.
Step 2: Isolate the ratio.
Dividing by $R_0$, \[ \frac{R}{R_0} = A^{1/3}. \]
Step 3: Take logarithms.
\[ \ln\!\left(\frac{R}{R_0}\right) = \ln\!\left(A^{1/3}\right). \]
Step 4: Use the power rule of logs.
\[ \ln\!\left(\frac{R}{R_0}\right) = \frac{1}{3}\ln A. \]
Step 5: Compare with a line.
Writing $y = \ln(R/R_0)$ and $x = \ln A$ gives $y = \dfrac{1}{3}x$, which is of the form $y = mx$, a straight line through the origin with slope $\dfrac13$.
Step 6: Conclude.
Therefore the graph is a straight line.
\[ \boxed{\text{A straight line}} \]