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A particle undergoes SHM with a time period of 2 seconds. In how much time will it travel from its mean position to a displacement equal to half of its amplitude :
A horizontal rod of mass m and length L is pivoted smoothly at one end. The rod ’s other end is supported by a spring of force constant k. The rod is rotated (in vertical plane) by a small angle q from its horizontal equilibrium position and released. The angular frequency of the
subsequent simple harmonic motion is :
it is not taken in calculation)
A traveling wave y = A sin passes from a heavier string to a lighter string. The reflected wave has amplitude 0.5 A. The junction of the strings is at x = 0. The equation of the reflected wave is:
As wave has been reflected from a rarer medium, therefore there is no change in phase. Hence equation for the
opposite direction can be written as
A string of length 1.5 m with its two ends clamped is vibrating in fundamental mode. Amplitude at the centre of the string is 4 mm. Minimum distance between the two points having amplitude 2 mm is:
The average density of Earth’s crust 10 km beneath the surface is 2.7 gm/cm^{3}. The speed of longitudnal seismic waves at that depth is 5.4 km/s. The bulk modulus of Earth’s crust considering its behavior as fluid at that depth, is :
The second overtone of an open pipe A and a closed pipe B have the same frequencies at a given temperature. Both pipes contain air. The ratio of fundamental frequency of A to the fundamental frequency of B is:
A thin uniform rod is suspended in vertical plane as a physical pendulum about point A. The time period of oscillation is T_{o}. Not counting the point A, the number 'n' of other points of suspension on rod such that the time period of oscillation (in vertical plane) is again T_{o}. Then the value of n is : (Since the rod is thin, consider one point for each transverse cross section of rod)
When the point of suspension is at a distance x from centre of length of rod, the time period of oscillation is
The time period of oscillation will be same (T^{0}) if the point of suspension
is a distance x = λ/2 or x = λ/6 from centre of the rod. Thus there will be three additional points.
Two radio station that are 250m apart emit radio waves of wavelength 100m. Point A is 400m from both station. Point B is 450m from both station. Point C is 400m from one station and 450 m from the other. The radio station emit radio waves in phase. Which of the following statement is true ?
Which of the following wave(s) can produce standing wave when superposed with y = Asin (ωt + kx)?
For standing wave to be formed the interfering waves must have same amplitude, same frequency and opposite direction of traveling. These are satisfied by options A, B and C.
A spring block system is put into SHM in two experiments. In the first, the block is pulled from the equilibrium position through a displacement d_{1} and then released. In the second, it is pulled from the equilibrium position through a greater distance d_{2} and then released. In both the experiments:
In Resonance tube experiment, if 400 Hz tuning fork is used, the first resonance occurs when length of air column is 19 cm. If the 400 Hz tuning fork is replaced by 1600 Hz tuning fork then to get resonance, the water level in the tube should be further lowered by (take end correction = 1 cm) :
Reasoning Type
Statement1 : A SHM may be assumed as composition of many SHM's.
Statement2 : Superposition of many SHM's (along same line) of same frequency will be a SHM.
Statement2 itself explains statement1.
REASONING TYPE
Statement1 : When a wave enters from one medium to another, its frequency is not changed.
Statement2 : Speed of a wave in a medium is property of the source.
Speed of wave in a medium is property of the medium.
A sinusoidal wave is propagating in negative x–direction in a string stretched along xaxis. A particle of string at x = 2m is found at its mean position and it is moving in positive y direction at t = 1 sec. The amplitude of the wave, the wavelength and the angular frequency of the wave are 0.1meter, π/4 meter and 4π rad/sec respectively.
Q. The equation of the wave is :
The equation of wave moving in negative xdirection, assuming origin of position at x = 2 and origin of time (i.e. initial time) at t = 1 sec.
Shifting the origin of position to left by 2m, that is, to x = 0. Also shifting the origin of time backwards by 1 sec, that
is to t = 0 sec.
A sinusoidal wave is propagating in negative x–direction in a string stretched along xaxis. A particle of string at x = 2m is found at its mean position and it is moving in positive y direction at t = 1 sec. The amplitude of the wave, the wavelength and the angular frequency of the wave are 0.1meter, π/4 meter and 4π rad/sec respectively.
Q. The speed of particle at x = 2 m and t = 1sec is :
A sinusoidal wave is propagating in negative x–direction in a string stretched along xaxis. A particle of string at x = 2m is found at its mean position and it is moving in positive y direction at t = 1 sec. The amplitude of the wave, the wavelength and the angular frequency of the wave are 0.1meter, π/4 meter and 4π rad/sec respectively.
Q. The instantaneous power transfer through x=2 m and t= 1.125 sec is :
Hence at t = 1.125 sec, that is, at T/4
seconds after t = 1 second, the particle is at rest at extreme position. Hence
instantaneous power at x = 2 at t = 1.125 sec is zero.
Matrix  Match Type
For a particle executing SHM along a straight line, match the statements in columnI with statement in columnII. (Note that displacement given in columnI is to be measured from mean position).
which maybe a circle if ω = 1 and ellipse of ω 1.Accelerationdisplacement graph is straight and acceleration time graph is sinusoidal
Two particles P_{1} and P_{2} are performing SHM along the same line about the same mean position. Initially they are at their positive extreme position. If the time period of each particle is 12 sec and the difference of their amplitudes is 12 cm then find the minimum time in seconds after which the separation between the particles become 6 cm.
A weightless rigid rod with a small iron bob at the end is hinged at point A to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible string of length 20 cm, fixed at its mid point. The bob is displaced slightly, perpendicular to the plane of the rod and string. Find period of small oscillations of the system in the form second. and fill value of X. (g = 10 m/s^{2})
The bob will execute SHM about a stationary axis passing through AB. If its effective length is l' then
A straight line source of sound of length L = 10m, emitts a pulse of sound that travels radially outward from the source. What sound energy (in mW) is intercepted by an acoustic cylindrical detector of surface area 2.4cm^{2}, located at a perpendicular distance 7m from the source. The waves reach perpendicularly at the surface of the detector. The total power emitted by the source in the form of sound is 2.2 × 10^{4} W.
(Use π = 22/7)
Imagine a cylinder of radius 7m and length 10m. Intensity of sound at the surface of cylinder is same everywhere.
In the figure shown strings AB and BC have masses m and 2m respectively. Both are of same length l. Mass of each string is uniformly distributed on its length. The string is suspended vertically from the ceiling of a room. A small jerk wave pulse is given at the end 'C'. It goes up to upper end 'A' in time 't'. If m = 2 kg, l = , Then find the value of 't' in seconds
For the given reaction the correct reactivity order of Hatoms is :
Reactivity depends upon stability of free radicals generated at y,zx and w respectively.
In the given sequence of reactions which of the following is the correct structure of compound A.
Which of the following is not correctly ordered for resonance stability
Follow the rules for stability of resonating structures (structure with more number of pbonds is more stable).
Amongst the following compounds select the strongest acid.
In acid (C), the group at ortho position is a –M group.
Amongst the following compounds select the strongest acid.
Sulphonic acid will be strongest acid.
Amongst the following compounds select the correct acidic strength order
–M effect increases acidic strength more as compare to –I effect.
Arrange the following carbocations in increasing order of stability:
Give the stability order of the following carbocations.
In (IV) and (I) the carbocation is directly stabilised by + m effect of NH_{2} and OCH_{3}
According to the rules.
Which of the following orders is/are correct with their mentioned property?
Ortho effect in first compound of (A) increases acidity.
In (C) N atom in first compound does not undergo resonance. Hence, more basic.
In (D) NH_{2}–OH is less basic because of –I effect of OH group.
In (B) first compound is more basic as lone pair of nitrogen easily available for the donation because of compact
structure.
In (C) basic strength in water is determined also due to Hbonding and solvation effect.
REASONING TYPE
Statement1 : Sulphanilic acid (pAminobenzene sulphonic acid) is stronger acid than sulphonic acid.
Statement2 : Sulphanilic acid exists as zwitter ion and ion is lost from group .
Statement1 : ohydroxybenzaldehyde is a liquid at room temperature while phydroxybenzaldehyde is a high melting solid.
Statement2 : Chelation make the former molecule thermodynamically more stable.
Intramolecular H bonding decreases its assocation with other molecules, hence it is a liquid.
The concept of resonance explains various properties of compounds. The molecules with conjugated system of π bonds, are stabilized by resonace and have low heat of hydrogenation. Hyperconjugative stabilization also decreases heat of hydrogenation. In aromatic rings a functional group with a lone pair of electron exerts + m effect. Some functional groups like –NO, –NC, –CH=CH_{2} can function both as electron releasing (+m, +R) or electron withdrawing (– m, – R) groups. More extended conjugation provides more stabilization.
Q. The correct heat of hydrogenation order is
(p) 1, 3Pentadiene (q) 1, 3Butadiene
(r) 2, 3Dimethyl1, 3butadiene (s) Propadiene
The concept of resonance explains various properties of compounds. The molecules with conjugated system of π bonds, are stabilized by resonace and have low heat of hydrogenation. Hyperconjugative stabilization also decreases heat of hydrogenation. In aromatic rings a functional group with a lone pair of electron exerts + m effect. Some functional groups like –NO, –NC, –CH=CH_{2} can function both as electron releasing (+m, +R) or electron withdrawing (– m, – R) groups. More extended conjugation provides more stabilization.
Q. The most stable carbocation is
The concept of resonance explains various properties of compounds. The molecules with conjugated system of π bonds, are stabilized by resonace and have low heat of hydrogenation. Hyperconjugative stabilization also decreases heat of hydrogenation. In aromatic rings a functional group with a lone pair of electron exerts + m effect. Some functional groups like –NO, –NC, –CH=CH_{2} can function both as electron releasing (+m, +R) or electron withdrawing (– m, – R) groups. More extended conjugation provides more stabilization.
Q. The most stable resonating structure of following compound is
Matrix  Match Type
Match the electronic effects responsible for the stability of given intermediates in ColumnI.
How many compounds give CO_{2} gas on reaction with NaHCO_{3} ?
Picric acid, benzoic acid and sulphonic acid give CO_{2} gas with NaHCO_{3}
In how many compounds delocalisation of electrons is possible ?
4, 6, 9, 10 have delocalisation.
How many products (structural isomers) are formed by monochlorination of given compound ?
All 13 carbon atoms are chemically different and all have hydrogen atoms.
Find the sum of number of stereoisomers and stereocentres for the given compound.
It is a case of symmetrical compound with 3 stereocenters & number of steroisomers = 2^{n–1} = 4.
So total = 7
In triangle ABC, area of triangle D and angle A are fixed. The angles B and C and length of sides a, b, c, are variable quantities. If length of the side a is minimum, then
Triangles ABC and DEF have sides of lengths a, b, c and d, e, f respectively (symbols are as per usual notations). a, b, c and d, e, f satisfy the relation then
Let θ_{1}, θ_{2}, θ_{3}, ..... be a sequence with θ_{1} = π/3_{ }and_{ }
If a, b, c are the sides of a triangle such that b c = λ^{2}, for some positive values of λ, then
If a, b, c are the sides of a triangle, then the minimum value of is
where n is a prime number greater than 2, then the value of
In a triangle ABC, a, b, and A are given, b > a and c_{1}, c_{2} are two possible values of the third side c. If Δ_{1} and Δ_{2} are areas of two triangles with sides a, b, c_{1} and a, b, c_{2} then
In a triangle ABC, points D and E are taken on side BC such that BD = DE = EC.
If ADE = AED = θ then
In a triangle ABC, which of the following is not possible ?
A triangle is inscribed in a circle, the vertex of triangle divides the circle into three arcs of length 3, 4 and 5
unit,then area of ΔABC is –
REASONING TYPE
Statement1 : In a ΔABC if cos A cos B + sin A sin B sin C = 1, then ABC is right angle triangle.
Statement2 : If k < p + qr < p + q < k, then k = p + qr = p + q = k and r = 1.
Let ABC be a triangle and D, E, F be the feet of perpendiculars from incentre to sides BC, CA, AB respectively. If R_{1}, R_{2}, R_{3} are radii of circles with centres C_{1}, C_{2}, C_{3} inscribed in quadrilaterals AFIE, BDIF, CEID respectively. Tangent M1N1 to circle with centre C_{1} is parallel to BC, where M_{1}N_{1} = x as shown in the figure. I_{1}, I_{2}, I_{3} are excentres of ΔABC.
Q. AC_{1} , AI, AI_{1} are proportional to
Let ABC be a triangle and D, E, F be the feet of perpendiculars from incentre to sides BC, CA, AB respectively. If R_{1}, R_{2}, R_{3} are radii of circles with centres C_{1}, C_{2}, C_{3} inscribed in quadrilaterals AFIE, BDIF, CEID respectively. Tangent M_{1}N_{1} to circle with centre C_{1} is parallel to BC, where M_{1}N_{1} = x as shown in the figure. I_{1}, I_{2}, I_{3} are excentres of ΔABC.
Q.
Let ABC be a triangle and D, E, F be the feet of perpendiculars from incentre to sides BC, CA, AB respectively. If R_{1}, R_{2}, R_{3} are radii of circles with centres C_{1}, C_{2}, C_{3} inscribed in quadrilaterals AFIE, BDIF, CEID respectively. Tangent M_{1}N_{1} to circle with centre C_{1} is parallel to BC, where M_{1}N_{1} = x as shown in the figure. I_{1}, I_{2}, I_{3} are excentres of ΔABC.
Q
A triangle ABC satisfies the relation 2sec4C + sin^{2}2A + = 0 and a point P is taken on the longest
side of the triangle such that it divides the side in the ratio 1 : 3. Let Q and R be the circumcenter and
orthocenter of ΔABC. If PQ : QR : RP = 1 : α : β, then find the value of α^{2} + β^{2}.
Find the smallest positive integer 'p' for which the equation cos (p sinx) = sin (p cosx) has a solution
in [0, 2π].
Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and x + y = 1.
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