Physics Question: Center of Mass of a Rod

Here are the study notes for the text:

• Physics Section*

Question 1: Centre of Mass

• A rod of length l has non-uniform linear mass density ρ(x) = a + b(x/l), where a and b are constants and 0 ≤ x ≤ l.
• The value of x for the centre of mass of the rod is at 3/2 (2a+b)l/(2a+3b)l.

Question 2: Angular Velocity

• A particle has an angular velocity of 10π rad s^{-1} at the end of 2 s and increases with constant angular acceleration to an angular velocity of 15π rad s^{-1} at the end of 4 s.
• The angular acceleration of the particle is 2.5π rad s^{-2}.

Question 3: Concave Mirror

• A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm in such a way that its end closer to the pole is 20 cm away from the mirror.
• The length of the image is 2.5 cm.

Question 4: Wavelengths

• If λ_v, λ_x, and λ_m represent the wavelengths of visible light, X-rays, and microwaves, respectively, then λ_m > λ_x > λ_v.

Question 5: Vertical Component of Velocity

• A body is projected horizontally from the top of a tower with an initial velocity of 18 m s^{-1}.
• It hits the ground at an angle of 45°.
• The vertical component of velocity when it strikes the ground is 9 m s^{-1}.

Question 6: Inductors in Parallel

• Two inductors L_1 and L_2 are connected in parallel and a time-varying current i flows as shown in the figure.
• The ratio of current i_1 to current i_2 at any time t is L_2/L_1.

Question 7: Dimensions of GIM^2

• If E = energy, G = gravitational constant, I = impulse, and M = mass, then the dimensions of GIM^2 are the same as those of E.

Question 8: Semiconductor Conductivity

• At room temperature, the electrical conductivity of a semiconductor is less than that of a good conductor.

Question 9: Transverse Wave

• The linear density of a vibrating string is 10^{-4} kg m^{-1}.
• A transverse wave is propagating on the string, which is described by the equation y = 0.02 sin(x + 30t), where x and y are in meters and the time t is in seconds.
• The tension in the string is 0.36 N.

Question 10: Work Done on an Electric Dipole

• An electric dipole of moment p is placed in a uniform electric field E with p parallel to E.
• The work done to rotate the dipole through an angle θ with the electric field is pE (1 - cosθ).

Question 11: Centre of Mass of a Stone and a Boy

• A stone of mass 500 g is dropped from the top of a tower of height 100 m and simultaneously another stone of mass 1 kg is thrown horizontally with a speed of 10 m s^{-1} from the same point.
• The height of the center of mass of the two-stone system after 3 s is 45 m.

Question 12: Change in Length of Wire

• One end of a wire 2 m long and diameter 2 mm is fixed in a ceiling.
• A naughty boy of mass 10 kg jumps to catch the free end and stays there.
• The change in length of the wire is 31.85 × 10^{-5} m.

Question 13: Additional Force Applied By the Engine

• An open carriage in a goods train is moving with a uniform velocity of 10 m s^{-1}.
• If the rain adds water with zero velocity at the rate of 5 kg s^{-1}, then the additional force applied by the engine to maintain the same velocity of the train is 0.5 N.

Question 14: Young's Double Slit Experiment

• In Young's double slit experiment, the source is white light.
• One slit is covered with red filter and the other with blue filter.
• There shall be dark fringes on the screen.

Question 15: Laser Beam Reflected by a Mirror

• The moon is 384000 km away from Earth.
• If a laser beam is sent to the moon and reflected back to the Earth by a mirror placed on the moon by an astronaut, the time taken by the light to make the round trip is 2.5 s.
• Chemistry Section*

Question 1: Sodium Lamp

• A 100 W sodium lamp radiates energy uniformly in all directions.
• The lamp is located at the center of a large sphere that absorbs all the sodium light which is incident on it.
• The wavelength of the sodium light is 589 nm.
• At what rate are the photons from the lamp associated with the sodium light delivered?

Question 2: Tyndall Effect

• The Tyndall effect can be observed in colloidal solutions.

Question 3: Reactivity of Metals with Water

• The reactivity of metals with water is in the order of Na > Mg > Zn > Fe > Cu.

Question 4: Paramagnetic Ion

• The ion that is paramagnetic is Cr^{3+}.

Question 5: Bohr's Radius

• The expression for Bohr's radius of an atom is r = (n^2 h^2) / (4π^2 m e^2 z^2).

Question 6: Oxidation Involves

• Oxidation involves loss of electrons.

Question 7: Silicon Dioxide

• In silicon dioxide, each silicon atom is surrounded by four oxygen atoms and each oxygen atom is bonded to two silicon atoms.

Question 8: Le-Chatelier's Principle

• Le-Chatelier's principle is applicable to a system in equilibrium.

Question 9: EMF of a Cell

• The standard EMF for the given cell reaction Zn + Cu^{2+} = Cu + Zn^{2+} is 1.10 V at 25°C.
• The EMF for the cell reaction, when 0.1M Cu^{2+} and 0.1M Zn^{2+} solutions are used, at 25°C is 1.10 V.Here are the study notes for the text:
• Chemistry*
• The correct order of reaction with electrophilic reagent is II > III > I.
• Molecular shapes of SF4, CF4, and XeF4 are different, with 0, 1, and 2 lone pairs of electrons respectively.
• Aniline in a set of reactions yielded a product D, which has the structure C6H5CH2NH2.
• Mathematics*
• Limit of √(1+√2+x) - √3 as x approaches 2 is equal to 1/√3.
• In a geometric sequence, if x, y, z and x, 4, z form an arithmetic sequence, then y = 2.
• Coordinate Geometry*
• The line passing through (0, 1) and perpendicular to the line x - 2y + 11 = 0 is 2x - y + 1 = 0.
• Probability*
• In a group of 400 people, the probability that a selected person is a smoker and non-vegetarian, given that the person is suffering from a chest disorder, is 14/45.
• Differential Equations*
• The solution of the differential equation dy/dx = x + y/x is yx = kx.
• The value of the sum of i - 2 - 3i + 4 + ... up to 100 terms, where i = √(-1) is -50(1 - i).
• Trigonometry*
• The value of cos(π/7) + cos(2π/7) + ... + cos(6π/7) + cos(7π/7) is -1.
• Algebra*
• The function f(x) = x^2 - 4x + 3 is injective as well as surjective.
• The equation ax^2 + bx + c = 0 has real roots if b^2 - 4ac ≥ 0.
• Geometry*
• If AB is a chord of the circle x^2 + y^2 = r^2 subtending a right angle at the centre, then the locus of the centroid of ΔPAB as P moves on the circle is a circle.
• Statistics*
• The mean age of 25 teachers in a school is 40 years. If a teacher retires at the age of 60 years and a new teacher is appointed in his place, then the age of the newly appointed teacher is 30 years.
• English and Logical Reasoning*
• Dr. R.P. Singh goes to see a patient at his residence after every 3.30 hours. If he has already gone 1.20 hours ago, then the information was given to the compounder by the doctor at 10.10 a.m.
• The odd one out among Nephrology, Entomology, Astrology, and Mycology is Astrology.
• Ashok said, "This girl is the wife of the grandson of my mother." So, Ashok is the grandfather of the girl's husband.

Let me know if you'd like me to clarify or expand on any of these points!### Code Problems

• H = 8 and HE = 13, so 'HEN' will be equal to 22.

Letter Patterns

• If O = 16 and FOR = 42, then FRONT will be equal to 61.

Number Patterns

• The odd one out is 28 - 49.

Alphabet Patterns

• ZX, XV, ?, T R, RP, so the answer is V U.

Analogies

• Distance : Odometer :: ? : Barometer, so the answer is Pressure.

Figure Counting

• There are 14 straight lines in the figure.

Code Problems

• 31 (CAGE) 75, so 89 (?) 45 is HIDE.

Figure Identification

• Find the identical figure from the given options.

Sequence Completion

• The correct sequence is given in option 1.

Idioms

• "Look down upon" means to consider someone inferior.
• "A close shave" means a narrow escape from danger.

Sentence Completion

• Stories depicting our culture and traditions have been handed down from generation to generation.
• The bail of cotton is very heavy.

Word Formation

• The word that cannot be formed using the letters of "Biological" is Bail.

Error Identification

• The error in the sentence is "takes" should be "take".

Word Meaning

• "Prophylactic" means preventive.

Sentence Completion

• If only you had spoken clearly, you would not have been misunderstood.

Antonyms

• The opposite of "Jettison" is Accept.

Word Meaning

• "Gesticulation" is a jumble or confused motion.