Physics 2009 Jamb Examination

As a ball-bearing falls through a viscous fluid, it initially accelerates due to gravity. As its velocity increases, viscous drag and upthrust also increase. Terminal velocity is achieved when the downward force of gravity equals the sum of the upward viscous and buoyant forces. At this point, the net force becomes zero, causing the ball to move with a constant or uniform velocity. Therefore, terminal velocity corresponds to the stage when velocity becomes uniform.

From the ideal gas equation:\[PV = nRT\]\[P = \dfrac{nRT}{V}\]\(n = 2\), \(R = 8.313\), \(T = 27 + 273 = 300\, K\), and \(V = 10^{-2}\).\[P = \dfrac{2 \times 8.313 \times 300}{10^{-2}} = \dfrac{4987.8}{10^{-2}}\]\[P = 4.99 \times 10^{5}\, \text{Nm}^{-2}\]Therefore, the pressure of the gas is 4.99 × 10⁵ Nm^-2.

From Newton’s law of gravitation:\[F = \dfrac{Gm_1m_2}{r^2}\]\[G = \dfrac{Fr^2}{m_1m_2}\]\(F\) has the dimension \(MLT^{-2}\).
Substituting gives:\[G = (MLT^{-2}) \times L^2 \times M^{-2}\]\[G = M^{-1}L^{3}T^{-2}\]\[\Rightarrow x = -1, \; y = 3, \; z = -2\]Thus, the dimensional expression is \(M^{-1}L^{3}T^{-2}\).

Adding a pentavalent impurity (such as phosphorus or arsenic) introduces extra electrons into a semiconductor lattice. These additional electrons act as majority charge carriers, enhancing conductivity. Such doping produces an n-type semiconductor because the current is mainly carried by negatively charged electrons rather than holes.

X-rays are a form of high-energy electromagnetic radiation that can penetrate deep into tissues. They are used in radiotherapy to destroy cancerous cells by damaging their DNA and disrupting their ability to reproduce. Healthy cells can repair themselves more effectively, which makes targeted X-ray treatment effective in cancer management.

The relationship between half-life \(t_{1/2}\) and decay constant \(\lambda\) is:\[t_{1/2} = \dfrac{0.693}{\lambda}\]\[t_{1/2} = \dfrac{0.693}{0.231} = 3.00\,\text{s}\]Hence, the half-life of the substance is 3.00 seconds.

The activity \(A\) of a radioactive sample is given by \(A = \lambda N\), where \(\lambda\) is the decay constant.
Half-life \(t_{1/2} = 20\,s\), thus:\[\lambda = \dfrac{0.693}{t_{1/2}} = \dfrac{0.693}{20} = 0.03465\,\text{s}^{-1}\]\(N = 10^{20}\). Therefore:\[A = 0.03465 \times 10^{20} = 3.47 \times 10^{18}\, \text{disintegrations per second}\]Hence, there are approximately 3.47 × 10¹⁸ disintegrations in the first second.

Thermionic emission occurs when a metal is heated to a high temperature, providing sufficient energy for electrons to overcome the work function and escape from the surface. This phenomenon is the basis of vacuum tubes and cathode-ray tubes, where electrons emitted from a hot cathode are accelerated toward an anode. It is purely thermal and does not involve nuclear or chemical reactions.

Copper is a metal with a large number of free mobile electrons in its conduction band. These electrons move freely when a potential difference is applied, allowing copper to conduct electricity efficiently. In contrast, germanium and silicon are semiconductors with limited conduction at room temperature, and grapefruit is an insulator.

The given wave equation is \( y = 10 \sin (1000\pi t - \pi x / 34) \).
Comparing with the general form \( y = A \sin (\omega t - kx) \):
\( k = \pi / 34 \).
The phase difference \( \Delta \phi \) between two points separated by a distance \( \Delta x \) is given by:\[ \Delta \phi = k \Delta x \] Substituting the values:\[ \Delta \phi = \dfrac{\pi}{34} \times 153 = 4.5\pi \; \text{radians} \] Converting radians to degrees:\[ 4.5\pi \times \dfrac{180}{\pi} = 810^\circ \] Since phase is periodic, \( 810^\circ \) is equivalent to \( 810 - 720 = 90^\circ \).
Therefore, the phase difference is 90°.