A torque \[\vec{\tau }\] on a body about a given point is found to be equal to \[\vec{C}\times \vec{L},\] where \[\vec{C}\] is a constant vector and \[\vec{L}\] is the angular momentum of the body about the point. From this, study the following statements:
(i) \[\vec{L}\] does not change with time
(ii) \[\frac{\overset{\to }{\mathop{dL}}\,}{dt}\] is perpendicular to \[\vec{L}\] at all instants of time
(iii) The magnitude of \[\vec{L}\] does not change with time
(iv) \[\frac{d\overset{\to }{\mathop{L}}\,}{dt}\] is parallel to \[\vec{L}\] at all instants of time
The velocity of a boat with respect to water is \[5\text{ }km/h\]making \[30{}^\circ \] with y-axis. The flow of river is\[5\text{ }km/hr\]. If the width of river is \[\left( \frac{\sqrt{3}}{2} \right)km\]as shown in figure, the coordinate of point where the boat reaches an opposite bank is
A uniform rope of mass 'm' is placed on a smooth fixed sphere of radius 'R' as shown in the figure. Find the tension T in the rope when it is in a horizontal plane at a vertical distance \[h=R/2\]below the top of sphere.
A cylinder with radius R spins about its horizontal axis with angular speed \[\omega \]. There is a small block lying on the inner surface of the cylinder. The coefficient of friction between the block and the cylinder is \[\mu \]. Find the value of co for which the block does not slip, i.e., stays at rest with respect to the cylinder.
A uniform dense rod with non-uniform Young's modulus is hanging from ceiling under gravity. If elastic energy density at every point is same, then Young's modulus with x will change in which of the shown graph.
A gas undergoes a cyclic process \[a-b-c-a\]which is as shown in the PV diagram. The process \[a-b\]is isothermal, \[b-c\]is adiabatic and \[c-a\]is a straight line on \[P-V\]diagram. Work done in process ab and be is \[5\text{ }J\]and \[4\text{ }J\]respectively. Calculate the efficiency of the cycle, if the area enclosed by the diagram abca in the figure is \[\text{3 }J\]
In figure-1 shown, \[{{m}_{1}}\] vertically oscillates with angular frequency\[{{\omega }_{1}}\]. Now the system is inverted so that \[{{m}_{2}}\] is at top it oscillated vertically it oscillates with angular frequency\[{{\omega }_{2}}\]. If it is kept on smooth ground as shown in figure-2, new angular frequency is
In the figure shown, initially the spring of negligible mass is in undeformed state and the block has zero velocity, E is a uniform electric field. Then
(i) The maximum speed of the block will be\[\frac{QE}{\sqrt{mK}}\]
(ii) The maximum speed of the block will be \[\frac{2QE}{\sqrt{mK}}\].
(iii) The maximum compression of the spring will be -\[\frac{QE}{K}\].
(iv) The maximum compression of the spring will be \[\frac{2QE}{K}\].
A short wire AB carrying \[{{I}_{1}}\] current lies in the plane of long wire which carry current I upward. If wire AB is released from horizontal position, and \[{{a}_{A}}\] and \[{{a}_{B}}\]are magnitude of acceleration of points A and B respectively correct alternative. (The space is gravity free).
Two point monochromatic and coherent sources of light of wavelength \[\lambda \] are placed on the dotted line in front of an infinite screen. The source emit waves in phase with each other. The distance between \[{{S}_{1}}\] and \[{{S}_{2}}\] is d while their distance from the screen is much larger. Choose the incorrect option.
A)
lf d is \[\frac{3\lambda }{2}\] at O, minima will be observed.
doneclear
B)
If d is \[\frac{11\lambda }{6},\] then intensity at O will be \[\frac{3}{4}\] of maximum intensity.
doneclear
C)
If d is \[3\lambda ,\] \[O\] will be a maxima.
doneclear
D)
If d is \[\frac{7\lambda }{6},\] the intensity at O will be equal to maximum intensity.
Two long straight conducting wires with linear mass density X are kept parallel to each other on a smooth horizontal surface. Distance between them is d and one end of each wire is connected to each other using a loose wire as shown in the figure. A capacitor is charged so as to have energy \[{{U}_{0}}\] stored in it. The capacitor is connected to the ends of two wires as shown. The resistance (R) of the entire arrangement is negligible and the capacitor discharges quickly. Assume that the distance between the wires do not change during the discharging process. Calculate the speed acquired by two wires as the capacitor discharges.
A non-conducting disc of radius R is uniformly charged with surface charge density \[\sigma \]. A disc of radius \[\frac{R}{2}\] is cut from the disc as shown in the figure. The electric potential at centre C of large disc will be
A glass capillary tube sealed at the upper end has internal radius r. The tube is held vertical with its lower end touching the surface of water. Calculate the length (L) of such a tube for water in it to rise to a height \[h(<L)\]. Atmospheric pressure is \[{{P}_{0}}\]and surface tension of water is T. Assume that water perfectly wets glass (Density of water \[=\rho \])
Two parallel plate capacitors of equal plate area are connected as shown in the figure. Initially each capacitor is charged to charge Q and separation between the plates is d. One plate of each capacitor moves with constant speed v keeping the other plate of capacitor remain fixed, as indicated in the figure. The current developed in the circuit is
Six identical conducting rods each of resistance R are connected in such a way that they form a regular triangular pyramid as shown in the figure. A battery is connected across A and C. Which of the following is correct?
A hemispherical shell of radius R is placed in a uniform electric field E as shown in figure. If \[\theta =45{}^\circ \] the electric flux through the hemispherical shell is
Two identical metallic small spheres of radius R are placed at very large separation and they are connected by a coil of inductance L having negligible resistance as shown in the figure. One of the sphere is given charge Q and then switch S is closed. The maximum current through the coil will be
A wheel of radius R rolls without slipping on the ground with a uniform velocity v. The relative acceleration of the topmost point of the wheel with respect to the bottommost point is
In figure (a), (b) a stationary spacecraft of mass M is passed by asteroid A of mass m, asteroid B of the same mass w, and asteroid C of mass 1m. The asteroids move along the indicated straight paths at the same speed; the perpendicular distances between the spacecraft on the paths are given as multiples of R. Figure gives the gravitational potential energy U(f) of the spacecraft- asteroid system during the passage of each asteroid treating time \[t=0\] as the moment when separation is minimum. Which asteroid corresponds to which plot of U(t)'?
The figure shows a plot of photocurrent v/s anode potential for a photo sensitive surface for three different radiations. Which one of the following is a correct statement?
A)
Curves and represent incident radiations of same frequency having same intensity
doneclear
B)
Curves and represent incident radiations of different frequencies and different intensities
doneclear
C)
Curves and represent incident radiations of same frequency but of different intensities
doneclear
D)
Curves and represent incident radiations of different frequencies and different intensities
A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40 amu) is kept at 300 K in a container. What is the ratio of the rms speeds \[\left( \frac{{{v}_{rms}}(helium)}{{{v}_{rms}}(\text{argon})} \right)\]?
A wire under tension vibrates with a fundamental frequency of\[600\text{ }Hz\]. If the length of the wire is doubled, the radius is halved and the wire is made to vibrate under one-ninth the tension. What is the fundamental frequency (in Hz)?
A convex lens of focal length 20 cm and another plano-convex lens of focal length 40 cm are placed co-axially. The plane surface of the plano-convex lens is silvered. An object O is kept on the principal axis at a distance of 10 cm from the convex lens. Find the distance d (in cm) between the two lenses so that final image is formed on the object itself.
A particle of mass \[2m\] is projected at an angle of \[45{}^\circ \] with horizontal with a velocity of \[20\sqrt{2}m/s\] . After 1 s explosion takes place and the particle is broken into two equal pieces. As a result of explosion one part comes to rest. What is the maximum height (in m) from the ground attained by the other part? \[(g=10m/{{s}^{2}})\]
If the following half cells have \[\operatorname{E}{}^\circ \] values as \[{{A}^{3+}}\,+{{e}^{-}}\xrightarrow{{}}{{A}^{2+}},\,\,E{}^\circ ={{y}_{2}}\,V\] \[{{A}^{2+}}\,+\,\,2{{e}^{-}}\xrightarrow{{}}A,\,\,E{}^\circ =-{{y}_{1}}\,V\] The \[E{}^\circ \] of the half-cell \[{{A}^{3+}}+3e\,\,\xrightarrow{{}}\,A\] will be
What is the elevation in boiling point of a solution of 13.44 g of \[{{\operatorname{CuCl}}_{2}}\] in 1 kg of water where molecular weight of \[{{\operatorname{CuCl}}_{2}}\] is 134.4 g and\[{{K}_{b}}\]\[0.52 K kg mo{{l}^{-}}^{1}\].
The densities of graphite and diamond at 298 K are 2.25 and \[3.31 g cm{{\,}^{-3}}\], respectively. If the standard free energy difference \[(\Delta G{}^\circ )\] is equal to \[1895 J mol{{\,}^{-}}^{1}\], the pressure at which graphite will be transformed into diamond at 298 K is
Among \[{{\operatorname{NH}}_{3}},\,\,HN{{O}_{3}},\,\,Na{{N}_{3}}\,\,and\,\,M{{g}_{3}}{{N}_{2}}\] what are the number of molecules having nitrogen in negative oxidation state?
A substance ?A? decomposes by a first order reaction starting initially with \[\left[ A \right] = 2.00 M\] and after 200 min, [A] becomes 0.15 M. What will be the half-life \[\,({{t}_{1/2}})\] of for this reaction in minutes?
If a system of the equation \[{{(\alpha +1)}^{3}}x+{{(\alpha +2)}^{3}}y-{{(\alpha +3)}^{3}}z=0,\] \[(\alpha +1)x+(\alpha +2)y-(\alpha +3)z=0\]and\[x+y-z=0\]x + y - z = 0 has infinitely many solutions, then what is the value of a?
If the standard deviation of the observations -5, -\[4,-3,-2,-1,0,1,2,3,4,5\]is \[\sqrt{10}\]. The standard deviation of observations \[15,16,17,18,19,20,21,\]\[22,23,24,25\] will be
The angle between the pair of lines \[({{x}^{2}}+{{y}^{2}})\left( \frac{{{\cos }^{2}}\theta }{4}+{{\sin }^{2}}\theta \right)={{\left( \frac{x}{\sqrt{3}}-y\sin \theta \right)}^{2}}\]is
where [x] and {x} denotes greatest integer and fractional part function respectively. The equation of the circles touching the line \[x+2y=0\]and passing through the points of intersection of the circle \[{{x}^{2}}+{{y}^{2}}=4\]and \[{{x}^{2}}+{{y}^{2}}-2x-4y+4=0\] is
If \[\alpha \]and \[\beta \]are the roots of \[6{{x}^{2}}-6x+1=0\], then the value of \[\frac{1}{2}[a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}]+\frac{1}{2}[a+b\beta +c{{\beta }^{2}}+d{{\beta }^{3}}]\]
If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}+12{{a}^{2}}x+1,\] \[a>0\]and f (x) is max. and min. at p and q respectively such that \[{{p}^{2}}=q\] then a =
The probability that Ram will alive 30 years hence is\[\frac{7}{11}\] and Shyam will be alive is \[\frac{7}{10}\]. What is the probability that both Ram and Shyam will be dead 30 years hence?
The number of distinct real values of X, for which the vectors \[-{{\lambda }^{2}}\hat{i}+\hat{j}+\hat{k},\hat{i}-{{\lambda }^{2}}\hat{j}+\hat{k}\]and \[\hat{i}+\hat{j}-{{\lambda }^{2}}\hat{k}\]are coplanar, is