Physics 2013 Jamb Examination

Mass is a fundamental property of an object that represents the quantity of matter it contains. Unlike weight, which is the force exerted on an object by gravity and varies depending on the gravitational field strength (\(W = mg\)), mass is an intrinsic property.
The gravitational pull on the moon is approximately one-sixth of that on Earth, so the brick's weight would decrease significantly. however, the actual amount of matter (atoms and molecules) making up the brick does not change simply by moving it to a different location in space. Therefore, the mass of the brick stays the same regardless of whether it is on the Earth, the moon, or in deep space.

When two forces are perpendicular to each other, they form the adjacent and opposite sides of a right-angled triangle, where the resultant force is the hypotenuse. We can use trigonometric ratios to find the magnitude of the individual force components.
Let the resultant force be \(R = 50\text{ N}\) and the angle it makes with one of the forces (let's call it \(F_1\)) be \(\theta = 30^\circ\). Using the cosine relationship for the adjacent side: \[ F_1 = R \cos \theta \]\[ F_1 = 50 \times \cos 30^\circ \]\[ F_1 = 50 \times 0.8660 \]\[ F_1 = 43.30\text{ N} \]Thus, the magnitude of the force that makes an angle of 30º with the resultant is 43.3N.

Scalar quantities are physical quantities that have magnitude only and no specific direction. Vector quantities, on the other hand, have both magnitude and direction.
Evaluating the options:
1. Volume and Area: Both describe size or extent without a required direction. These are scalars.
2. Moment and Momentum: Both are vector quantities as they depend on the direction of force or velocity.
3. Length and Displacement: Length is a scalar, but displacement is a vector (distance in a specified direction).
4. Impulse and Time: Time is a scalar, but impulse is a vector (change in momentum).
Therefore, the only pair where both quantities are strictly scalars is volume and area.

The period \(T\) of a simple pendulum is related to its length \(l\) by the formula:\[ T = 2\pi \sqrt{\frac{l}{g}} \]Since the pendulums are at the same place, \(g\) is constant. This implies that the period is directly proportional to the square root of the length:\[ T \propto \sqrt{l} \quad \Rightarrow \quad \frac{T_1}{\sqrt{l_1}} = \frac{T_2}{\sqrt{l_2}} \]Given:
\(l_1 = 0.4\text{ m}\)
\(T_1 = 2\text{ s}\)
\(l_2 = 0.8\text{ m}\)
Substituting the values:\[ \frac{2}{\sqrt{0.4}} = \frac{T_2}{\sqrt{0.8}} \]\[ T_2 = 2 \times \sqrt{\frac{0.8}{0.4}} \]\[ T_2 = 2 \times \sqrt{2} \]\[ T_2 = 2\sqrt{2}\text{ s} \]

We can determine the time using the first equation of linear motion:\[ v = u + at \]Where:
\(u = 20\text{ ms}^{-1}\) (initial velocity)
\(v = 0\text{ ms}^{-1}\) (final velocity, as it comes to a halt)
\(a = -2\text{ ms}^{-2}\) (deceleration is negative acceleration)
\(t = \text{?}\)
Substituting the values into the equation:\[ 0 = 20 + (-2)t \]\[ 0 = 20 - 2t \]\[ 2t = 20 \]\[ t = \frac{20}{2} = 10\text{ s} \]The train requires 10 seconds to come to a complete stop.

When an elevator moves downwards with an acceleration \(a\), the reaction force \(R\) (the apparent weight) felt by the person is less than their true weight \(mg\). The equation of motion is:\[ mg - R = ma \]The apparent weight loss is the difference between the true weight and the apparent weight:\[ \text{Weight Loss} = mg - R \]From the equation above, we see that:\[ \text{Weight Loss} = ma \]Given:
\(m = 70\text{ kg}\)
\(a = 1.5\text{ ms}^{-2}\)\[ \text{Weight Loss} = 70 \times 1.5 \]\[ \text{Weight Loss} = 105\text{ N} \]The man experiences an apparent weight loss of 105N.

According to the principle of floatation, a floating object displaces its own weight of the liquid in which it floats. The fraction of the volume immersed is equal to the ratio of the density of the floating object to the density of the liquid.\[ \text{Fraction immersed} = \frac{\text{Density of cork}}{\text{Density of liquid}} \]\[ \text{Fraction immersed} = \frac{0.25 \times 10^3}{1.25 \times 10^3} \]\[ \text{Fraction immersed} = \frac{0.25}{1.25} \]\[ \text{Fraction immersed} = \frac{1}{5} = 0.2 \]Thus, 0.2 (or 20%) of the cork's volume is submerged.

We are looking for the height \(h\) where Potential Energy (P.E.) equals Kinetic Energy (K.E.).
The formulas are:\[ P.E. = mgh \]\[ K.E. = \frac{1}{2}mv^2 \]Setting them equal as per the question:\[ mgh = \frac{1}{2}mv^2 \]Since the bodies are 'similar', we assume the masses \(m\) are equal and they cancel out:\[ gh = \frac{1}{2}v^2 \]Given \(v = 5\text{ ms}^{-1}\) and assuming \(g = 10\text{ ms}^{-2}\) (standard for these problems unless otherwise stated):\[ 10 \times h = \frac{1}{2} \times 5^2 \]\[ 10h = \frac{1}{2} \times 25 \]\[ 10h = 12.5 \]\[ h = \frac{12.5}{10} = 1.25\text{ m} \]Rounding to the nearest decimal as provided in the options, we get 1.3m.

Power is defined as the rate of doing work. In this case, the work done is equal to the gravitational potential energy gained by the water.\[ \text{Power} = \frac{\text{Work Done}}{\text{Time}} = \frac{mgh}{t} \]Given:
\(m = 5000\text{ kg}\)
\(g = 10\text{ ms}^{-2}\)
\(h = 60\text{ m}\)
\(t = 15\text{ min} = 15 \times 60 = 900\text{ s}\)
Substituting the values:\[ P = \frac{5000 \times 10 \times 60}{900} \]\[ P = \frac{3,000,000}{900} \]\[ P = \frac{30,000}{9} \]\[ P = 3333.33\text{ Js}^{-1} \]Converting to scientific notation:\[ P \approx 3.3 \times 10^3\text{ Js}^{-1} \]

The work done in stretching or compressing a spring is given by the area under the Force-extension graph, which is a triangle since the force varies linearly with extension (Hooke's Law). The formula is:\[ W = \frac{1}{2}Fe \]Where:
\(F = 20\text{ N}\)
\(e = 50\text{ mm} = 0.05\text{ m}\) (converting to standard SI units)
Substituting the values:\[ W = \frac{1}{2} \times 20 \times 0.05 \]\[ W = 10 \times 0.05 \]\[ W = 0.5\text{ J} \]Therefore, the work done is 0.5 Joules.