Physics 2005 Jamb Examination

When many soldiers march in step the periodic force they apply can match a bridge's natural frequency. If that happens the small periodic forces add coherently and the amplitude of the bridge's oscillation grows — a phenomenon called resonance.

Resonance can produce very large oscillations that may damage or even collapse structures. Breaking step prevents the applied forcing from becoming periodic at the bridge's natural frequency, so resonant amplification is avoided.

The sharpness (definition) of a shadow depends on how much light is blocked versus transmitted and how coherent the light reaching the obstacle is. An opaque object blocks light cleanly and tends to produce a well-defined (sharp) umbra; partial transparency produces a fuzzier shadow.

In practice the size of the light source and the object's opacity both matter: a point source + opaque object → sharp shadow; extended source or semi-transparent object → penumbra and a blurred edge. Among the given choices, opacity best captures this effect.

Lens power \(P\) (in diopters) is the reciprocal of focal length \(f\) (in metres): \[P=\dfrac{1}{f}.\] For \(P=4.0\,\text{D}\) we have \[f=\dfrac{1}{P}=\dfrac{1}{4.0}=0.25\,\text{m}.\]

This is a straightforward application of the definition of optical power; a positive power indicates a converging lens and the focal length is 0.25 m (option D).

Let object distance be \(u\) and image distance \(v\). Magnification (linear) is \(m=\dfrac{v}{u}=3\) (real image, same direction convention by magnitude). Given the image is 24 cm from the mirror, take \(v=24\,\text{cm}\); thus \(u=\dfrac{v}{3}=8\,\text{cm}.\)

Use the mirror formula \[\dfrac{1}{f}=\dfrac{1}{u}+\dfrac{1}{v}=\dfrac{1}{8}+\dfrac{1}{24}=\dfrac{3+1}{24}=\dfrac{4}{24}=\dfrac{1}{6}.\] Hence \(f=6\,\text{cm}\) (option B). This matches the geometry: a relatively short focal length produces the magnified real image at the given distance.

Vector quantities have both magnitude and direction. Momentum and torque are vectors, but power and speed are scalars. Velocity and displacement are vectors while distance and speed are scalars. Acceleration is a vector but speed is a scalar.

Here displacement (a vector) and torque (a vector — technically a pseudovector) are both directional quantities, so option C is correct. (Torque is often treated as a vector in mechanics conventions.)

In a wheel and axle, the effort is applied at the wheel of radius \(R\) and the load is lifted at the axle of radius \(r\). The mechanical advantage (M.A.) is given by the ratio of the effort arm to the load arm, i.e. \(M.A.=\tfrac{R}{r}.\)

This result assumes an ideal system without friction. Squared terms would only appear if considering areas or moments of inertia, which is not the case here. Hence the correct expression is \(R/r\), option D.

for a body projected at an angle θ to the horizontal, the time of the flight is given by

\[T = \dfrac{2u sin θ}{g}\]

\[ = \dfrac{2 \times 40) \times sin (30)}{10}\]

\[ = {2 \times 40 \times (0.5)}{10} = 4.0 secs.\]

For a purely capacitive a.c circuit, the current leads the voltage by 90°. Equivalently, the voltage lags the current by 90°. This is a standard phase relationship in alternating current theory.

The reason is that capacitor voltage is proportional to charge, while current is the time derivative of charge. Differentiation introduces a phase lead of 90°, giving the observed phase shift.

Each half-life reduces the amount by a factor of 1/2. To reach 1/32 of the original, note that \(1/32=(1/2)^5.\) That means 5 half-lives are required.

Since each half-life is 50 days, the total time is \(5\times 50=250\,\text{days}.\) Correct option: A.

The magnetic force is \(F=qvB\sin\theta.\) Substituting: \(F=8.0\times10^{-14}, q=1.6\times10^{-19}, v=1.0\times10^6, B=1.0.\) Compute denominator: \(qvB=1.6\times10^{-19}\times10^6=1.6\times10^{-13}.\)

So, \(\sin\theta=\tfrac{8.0\times10^{-14}}{1.6\times10^{-13}}=0.5.\) Thus \(\theta=30^\circ\) or 150°. \