Match Column Parameters with Graph Representations in Physics Experiments

To match the parameters in Column I with the corresponding graph representations in Column II, let us analyze each statement mathematically:

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(A) Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis)

For small oscillations, the potential energy $U$ of a simple pendulum at a displacement $x$ from its equilibrium position is given by:
$U(x) = U_0 + \frac{1}{2} m \omega^2 x^2$
where $U_0 \ge 0$ is the potential energy at the mean position ($x = 0$), $m$ is the mass, and $\omega$ is the angular frequency.

• Graphical representation:
  Here, $y = U(x)$ and $x$ is the displacement. The relation is quadratic of the form $y = c_1 + c_2 x^2$ ($c_2 > 0$), which represents a parabola opening upwards with its minimum (vertex) located at a non-zero positive value on the $y$-axis ($y = U_0 > 0$).
• This uniquely matches graph (p).

$\mathbf{A \rightarrow p}$

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(B) Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration when the body is moving along the positive $x$-direction

The position or displacement $y$ as a function of time $t$ (here $x$-axis) is given by the equation of motion:
$y(x) = y_0 + v_0 x + \frac{1}{2} a x^2$

1. Zero acceleration ($a = 0$):

   • If the initial displacement $y_0 = 0$, then $y = v_0 x$, which is a straight line passing through the origin with a positive slope ($v_0 > 0$). This corresponds to graph (q).
   • If the initial displacement $y_0 > 0$, then $y = y_0 + v_0 x$, which is a straight line with a positive $y$-intercept and a positive slope. This corresponds to graph (r).

2. Constant positive acceleration ($a > 0$):

   • If initial position $y_0 = 0$ and initial velocity $v_0 = 0$, then $y = \frac{1}{2} a x^2$, which represents a parabola passing through the origin and opening upwards in the first quadrant. This corresponds to graph (s).

$\mathbf{B \rightarrow q, r, s}$

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(C) Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle

The horizontal range $R$ of a projectile launched with initial velocity $u$ at a fixed angle $\theta$ to the horizontal is:
$R = \frac{u^2 \sin 2\theta}{g}$

• Graphical representation:
  Let $y = R$ and $x = u$. The relationship becomes:
  $y = \left(\frac{\sin 2\theta}{g}\right) x^2 = C x^2 \quad (C > 0)$
  This is the equation of a parabola opening upwards and passing through the origin $(0,0)$.
• This corresponds to graph (s).

$\mathbf{C \rightarrow s}$

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(D) The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis)

The time period $T$ of a simple pendulum of length $L$ is given by:
$T = 2\pi \sqrt{\frac{L}{g}}$

Squaring both sides gives:
$T^2 = \left(\frac{4\pi^2}{g}\right) L$

• Graphical representation:
  Let $y = T^2$ and $x = L$. The relationship becomes:
  $y = \left(\frac{4\pi^2}{g}\right) x = m x \quad (m > 0)$
  This is a straight line passing through the origin $(0,0)$ with a positive slope.
• This corresponds to graph (q).

$\mathbf{D \rightarrow q}$

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Final Match:

• A $\rightarrow$ p
• B $\rightarrow$ q, r, s
• C $\rightarrow$ s
• D $\rightarrow$ q