General
Easy
Physics-

A piece of wire is bent in the shape of a parabola y equals k x to the power of 2 end exponent blank left parenthesis y-axis vertical) with a bead of mass m on it. The bead can side on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the x-axis with a constant acceleration a. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire, from the y-axis is

Physics-General

  1. a divided by g k    
  2. a divided by 2 g k    
  3. 2 a divided by g k    
  4. a divided by 4 g k    

    Answer:The correct answer is: a divided by 2 g km a cos invisible function application theta equals m g cos invisible function application left parenthesis 90 minus theta right parenthesis
    rightwards double arrow fraction numerator a over denominator g end fraction equals tan invisible function application theta rightwards double arrow fraction numerator a over denominator g end fraction equals fraction numerator d y over denominator d x end fraction
    rightwards double arrow fraction numerator d over denominator d x end fraction open parentheses k x close parentheses to the power of 2 end exponent equals fraction numerator a over denominator g end fraction rightwards double arrow x equals fraction numerator a over denominator 2 g k end fraction

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    General
    physics-

    A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length s equals t to the power of 3 end exponent plus 5 comma where s is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

    G i v e n comma blank s equals t to the power of 3 end exponent plus 5
    S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent
    a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t
    therefore T a n g e n t i a l blank a c c e l e r a t i o n blank a t blank t equals 2 blank s comma
    a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent
    a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent
    therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent
    therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent

    A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of P is such that it sweeps out length s equals t to the power of 3 end exponent plus 5 comma where s is in metre and t is in second. The radius of the path is 20 m. The acceleration of P when t =2s is nearly

    physics-General
    G i v e n comma blank s equals t to the power of 3 end exponent plus 5
    S p e e d comma blank v equals fraction numerator d s over denominator d t end fraction equals 3 t to the power of 2 end exponent
    a n d blank r a t e blank o f blank c h a n g e blank o f blank s p e e d comma blank a subscript t end subscript equals fraction numerator d v over denominator d t end fraction equals 6 t
    therefore T a n g e n t i a l blank a c c e l e r a t i o n blank a t blank t equals 2 blank s comma
    a subscript t end subscript equals 6 cross times 2 equals 12 blank m s to the power of negative 2 end exponent
    a n d blank a t blank t equals 2 s comma blank v equals 3 left parenthesis 2 right parenthesis to the power of 2 end exponent equals 12 m s to the power of negative 1 end exponent
    therefore C e n t r i p e t a l blank a c c e l e r a t i o n comma blank a subscript c end subscript equals fraction numerator v to the power of 2 end exponent over denominator R end fraction equals fraction numerator 144 over denominator 20 end fraction m s to the power of negative 2 end exponent
    therefore N e t blank a c c e l e r a t i o n equals a subscript t end subscript superscript 2 end superscript plus a subscript i end subscript superscript 2 end superscript almost equal to 14 m s to the power of negative 2 end exponent
    General
    maths-

    The equation of the plane containing the line fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text  is  end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0 where al + bm + cn is equal to

     

    Since these two lines are intersecting so shortest distance between the lines will be 0.
    Hence (c) is the correct answer.

    The equation of the plane containing the line fraction numerator x with not stretchy bar on top minus alpha over denominator 1 end fraction equals fraction numerator y minus beta over denominator m end fraction equals fraction numerator z minus gamma over denominator n end fraction text  is  end text stack a with _ below with _ below left parenthesis x minus alpha right parenthesis plus b left parenthesis y minus beta right parenthesis plus c left parenthesis z minus gamma right parenthesis equals 0 where al + bm + cn is equal to

     

    maths-General
    Since these two lines are intersecting so shortest distance between the lines will be 0.
    Hence (c) is the correct answer.
    General
    physics-

    A small body of mass m slides down from the top of a hemisphere of radius r. The surface of block and hemisphere are frictionless. The height at which the body lose contact with the surface of the sphere is

    A small body of mass m slides down from the top of a hemisphere of radius r. The surface of block and hemisphere are frictionless. The height at which the body lose contact with the surface of the sphere is

    physics-General
    General
    physics-

    Average torque on a projectile of mass m, initial speed u and angles of projection theta, between initial and final position P and Q as shown in figure about the point of projection is

    Time of flight. T equals fraction numerator 2 u sin invisible function application theta over denominator g end fraction
    Horizontal range, R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
    Change in angular momentum,
    open vertical bar d stack L with rightwards arrow on top close vertical bar equals open vertical bar stack L with rightwards arrow on top subscript f end subscript minus stack L with rightwards arrow on top subscript i end subscript close vertical bar about point of projection
    equals left parenthesis m u sin invisible function application theta right parenthesis cross times fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
    equals fraction numerator m u to the power of 3 end exponent sin invisible function application theta sin invisible function application 2 theta over denominator g end fraction
    T o r q u e blank open vertical bar stack tau with rightwards arrow on top close vertical bar equals fraction numerator c h a n g e blank i n blank a n g u l a r blank m o m e n t u m over denominator t i m e blank o f blank f l i g h t end fraction
    equals open vertical bar fraction numerator d stack L with rightwards arrow on top over denominator T end fraction close vertical bar

    Average torque on a projectile of mass m, initial speed u and angles of projection theta, between initial and final position P and Q as shown in figure about the point of projection is

    physics-General
    Time of flight. T equals fraction numerator 2 u sin invisible function application theta over denominator g end fraction
    Horizontal range, R equals fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
    Change in angular momentum,
    open vertical bar d stack L with rightwards arrow on top close vertical bar equals open vertical bar stack L with rightwards arrow on top subscript f end subscript minus stack L with rightwards arrow on top subscript i end subscript close vertical bar about point of projection
    equals left parenthesis m u sin invisible function application theta right parenthesis cross times fraction numerator u to the power of 2 end exponent sin invisible function application 2 theta over denominator g end fraction
    equals fraction numerator m u to the power of 3 end exponent sin invisible function application theta sin invisible function application 2 theta over denominator g end fraction
    T o r q u e blank open vertical bar stack tau with rightwards arrow on top close vertical bar equals fraction numerator c h a n g e blank i n blank a n g u l a r blank m o m e n t u m over denominator t i m e blank o f blank f l i g h t end fraction
    equals open vertical bar fraction numerator d stack L with rightwards arrow on top over denominator T end fraction close vertical bar
    General
    physics-

    Three balls are dropped from the top of a building with equal speed at different angles. When the balls strike ground their velocities are v subscript 1 end subscript comma v blank subscript 2 end subscript and v subscript 3 end subscript respectively, then

    All the balls are projected from the same height, therefore their velocities will be equal.
    S o comma blank v subscript 1 end subscript equals v subscript 2 end subscript equals v subscript 3 end subscript

    Three balls are dropped from the top of a building with equal speed at different angles. When the balls strike ground their velocities are v subscript 1 end subscript comma v blank subscript 2 end subscript and v subscript 3 end subscript respectively, then

    physics-General
    All the balls are projected from the same height, therefore their velocities will be equal.
    S o comma blank v subscript 1 end subscript equals v subscript 2 end subscript equals v subscript 3 end subscript
    General
    physics-

    A string of length L is fixed at one end and the string makes fraction numerator 2 over denominator pi end fraction rev/s around the vertical axis through, the fixed and as shown in the figure, then tension in the string is

    T sin invisible function application theta equals M omega to the power of 2 end exponent R (i)
    T sin invisible function application theta equals M omega to the power of 2 end exponent L blank s i n blank theta (ii)
    T equals M omega to the power of 2 end exponent L
    equals M bullet 4 pi to the power of 2 end exponent n to the power of 2 end exponent L
    equals M bullet 4 pi to the power of 2 end exponent open parentheses fraction numerator 2 over denominator pi end fraction close parentheses to the power of 2 end exponent L
    equals 16 blank M L

    A string of length L is fixed at one end and the string makes fraction numerator 2 over denominator pi end fraction rev/s around the vertical axis through, the fixed and as shown in the figure, then tension in the string is

    physics-General
    T sin invisible function application theta equals M omega to the power of 2 end exponent R (i)
    T sin invisible function application theta equals M omega to the power of 2 end exponent L blank s i n blank theta (ii)
    T equals M omega to the power of 2 end exponent L
    equals M bullet 4 pi to the power of 2 end exponent n to the power of 2 end exponent L
    equals M bullet 4 pi to the power of 2 end exponent open parentheses fraction numerator 2 over denominator pi end fraction close parentheses to the power of 2 end exponent L
    equals 16 blank M L
    General
    physics-

    A thin prism P subscript 1 end subscript with angle 6 to the power of ring operator end exponent and made from glass of refractive index 1.54 is combined with another thin prism P subscript 2 end subscript of refractive index 1.72 to produce dispersion without deviation. The angle of prism P subscript 2 end subscript will be

    fraction numerator A to the power of ´ end exponent over denominator A end fraction equals fraction numerator left parenthesis mu subscript y end subscript minus 1 right parenthesis over denominator left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis end fraction rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator 6 end fraction equals negative fraction numerator left parenthesis 1.54 minus 1 right parenthesis over denominator left parenthesis 1.72 minus 1 right parenthesis end fraction
    Þ A' equals negative 4.5 to the power of o end exponent equals 4 to the power of o end exponent 3 0 to the power of ´ end exponent

    A thin prism P subscript 1 end subscript with angle 6 to the power of ring operator end exponent and made from glass of refractive index 1.54 is combined with another thin prism P subscript 2 end subscript of refractive index 1.72 to produce dispersion without deviation. The angle of prism P subscript 2 end subscript will be

    physics-General
    fraction numerator A to the power of ´ end exponent over denominator A end fraction equals fraction numerator left parenthesis mu subscript y end subscript minus 1 right parenthesis over denominator left parenthesis mu subscript y ´ end subscript minus 1 right parenthesis end fraction rightwards double arrow fraction numerator A to the power of ´ end exponent over denominator 6 end fraction equals negative fraction numerator left parenthesis 1.54 minus 1 right parenthesis over denominator left parenthesis 1.72 minus 1 right parenthesis end fraction
    Þ A' equals negative 4.5 to the power of o end exponent equals 4 to the power of o end exponent 3 0 to the power of ´ end exponent
    General
    physics-

    A triangular prism of glass is shown in the figure. A ray incident normally to one face is totally reflected, if theta equals 4 5 to the power of o end exponent. The index of refraction of glass is

    For total internal reflection theta greater than C
    rightwards double arrow sin invisible function application theta greater than sin invisible function application C rightwards double arrow sin invisible function application theta greater than fraction numerator 1 over denominator mu end fraction
    or mu greater than fraction numerator 1 over denominator sin invisible function application theta end fraction rightwards double arrow mu greater than fraction numerator 1 over denominator sin invisible function application 4 5 to the power of o end exponent end fraction rightwards double arrow mu greater than square root of 2 rightwards double arrow mu greater than 1.41

    A triangular prism of glass is shown in the figure. A ray incident normally to one face is totally reflected, if theta equals 4 5 to the power of o end exponent. The index of refraction of glass is

    physics-General
    For total internal reflection theta greater than C
    rightwards double arrow sin invisible function application theta greater than sin invisible function application C rightwards double arrow sin invisible function application theta greater than fraction numerator 1 over denominator mu end fraction
    or mu greater than fraction numerator 1 over denominator sin invisible function application theta end fraction rightwards double arrow mu greater than fraction numerator 1 over denominator sin invisible function application 4 5 to the power of o end exponent end fraction rightwards double arrow mu greater than square root of 2 rightwards double arrow mu greater than 1.41
    General
    physics-

    Which of the following diagrams, shows correctly the dispersion of white light by a prism

    Because in dispersion of white light, the rays of different colours are not parallel to each other. Also deviation takes place in same direction.

    Which of the following diagrams, shows correctly the dispersion of white light by a prism

    physics-General
    Because in dispersion of white light, the rays of different colours are not parallel to each other. Also deviation takes place in same direction.
    General
    physics-

    A ray of light incident normally on an isosceles right angled prism travels as shown in the figure. The least value of the refractive index of the prism must be

    A ray of light incident normally on an isosceles right angled prism travels as shown in the figure. The least value of the refractive index of the prism must be

    physics-General
    General
    physics-

    A ray of monochromatic light is incident on one refracting face of a prism of angle 75 to the power of ring operator end exponent. It passes through the prism and is incident on the other face at the critical angle. If the refractive index of the material of the prism is square root of 2, the angle of incidence on the first face of the prism is

    From figure
    A equals r subscript 1 end subscript plus c equals r subscript 1 end subscript plus sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses
    rightwards double arrow r subscript 1 end subscript equals 75 minus sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses
    rightwards double arrow 75 minus 45 equals 3 0 to the power of o end exponent
    From Snell’s law At B
    mu equals fraction numerator sin invisible function application i over denominator sin invisible function application r subscript 1 end subscript end fraction rightwards double arrow square root of 2 equals fraction numerator sin invisible function application i over denominator sin invisible function application 3 0 to the power of o end exponent end fraction

    A ray of monochromatic light is incident on one refracting face of a prism of angle 75 to the power of ring operator end exponent. It passes through the prism and is incident on the other face at the critical angle. If the refractive index of the material of the prism is square root of 2, the angle of incidence on the first face of the prism is

    physics-General
    From figure
    A equals r subscript 1 end subscript plus c equals r subscript 1 end subscript plus sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses
    rightwards double arrow r subscript 1 end subscript equals 75 minus sin to the power of negative 1 end exponent invisible function application open parentheses fraction numerator 1 over denominator mu end fraction close parentheses
    rightwards double arrow 75 minus 45 equals 3 0 to the power of o end exponent
    From Snell’s law At B
    mu equals fraction numerator sin invisible function application i over denominator sin invisible function application r subscript 1 end subscript end fraction rightwards double arrow square root of 2 equals fraction numerator sin invisible function application i over denominator sin invisible function application 3 0 to the power of o end exponent end fraction
    General
    physics-

    A light ray is incident upon a prism in minimum deviation position and suffers a deviation of 34°. If the shaded half of the prism is knocked off, the ray will

    By formula delta equals left parenthesis n minus 1 right parenthesis A rightwards double arrow 34 equals left parenthesis n minus 1 right parenthesis A and in the second position delta to the power of ´ end exponent equals left parenthesis n minus 1 right parenthesis fraction numerator A over denominator 2 end fraction
    therefore fraction numerator 34 over denominator delta to the power of ´ end exponent end fraction equals fraction numerator left parenthesis n minus 1 right parenthesis A over denominator left parenthesis n minus 1 right parenthesis fraction numerator A over denominator 2 end fraction end fraction or delta to the power of ´ end exponent equals fraction numerator 34 over denominator 2 end fraction equals 1 7 to the power of o end exponent

    A light ray is incident upon a prism in minimum deviation position and suffers a deviation of 34°. If the shaded half of the prism is knocked off, the ray will

    physics-General
    By formula delta equals left parenthesis n minus 1 right parenthesis A rightwards double arrow 34 equals left parenthesis n minus 1 right parenthesis A and in the second position delta to the power of ´ end exponent equals left parenthesis n minus 1 right parenthesis fraction numerator A over denominator 2 end fraction
    therefore fraction numerator 34 over denominator delta to the power of ´ end exponent end fraction equals fraction numerator left parenthesis n minus 1 right parenthesis A over denominator left parenthesis n minus 1 right parenthesis fraction numerator A over denominator 2 end fraction end fraction or delta to the power of ´ end exponent equals fraction numerator 34 over denominator 2 end fraction equals 1 7 to the power of o end exponent
    General
    physics-

    A prism ABC of angle 30° has its face AC silvered. A ray of light incident at an angle of 45° at the face AB retraces its path after refraction at face AB and reflection at face AC. The refractive index of the material of the prism is

    A prism ABC of angle 30° has its face AC silvered. A ray of light incident at an angle of 45° at the face AB retraces its path after refraction at face AB and reflection at face AC. The refractive index of the material of the prism is

    physics-General
    General
    physics-

    A ray of light passes through an equilateral glass prism in such a manner that the angle of incidence is equal to the angle of emergence and each of these angles is equal to 3/4 of the angle of the prism. The angle of deviation is

    Is a band spectrum
    In the position of minimum deviation
    2 i equals A plus delta subscript m end subscript or delta subscript m end subscript equals 2 i minus A equals 90 minus 60 equals 3 0 to the power of o end exponent

    A ray of light passes through an equilateral glass prism in such a manner that the angle of incidence is equal to the angle of emergence and each of these angles is equal to 3/4 of the angle of the prism. The angle of deviation is

    physics-General
    Is a band spectrum
    In the position of minimum deviation
    2 i equals A plus delta subscript m end subscript or delta subscript m end subscript equals 2 i minus A equals 90 minus 60 equals 3 0 to the power of o end exponent
    General
    physics-

    A small particle of mass m blankis projected at an angle theta with the x-axis with an initial velocity v subscript 0 end subscript in the x-y plane as shown in the figure. A t blank a blank t i m e blank t less than fraction numerator v subscript 0 end subscript sin invisible function application theta over denominator g end fraction comma the angular momentum of the particle is

    L equals m open parentheses r cross times v close parentheses
    L equals m open square brackets v subscript 0 end subscript c o s theta blank t stack stack i with dot on top with hat on top plus open parentheses v subscript 0 end subscript s i n theta blank t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent close parentheses stack stack j with dot on top with hat on top close square brackets
    cross times open square brackets v subscript 0 end subscript c o s theta blank stack stack i with dot on top with hat on top plus open parentheses v subscript 0 end subscript s i n theta minus g t close parentheses stack stack j with dot on top with hat on top close square brackets
    equals m v subscript 0 end subscript c o s theta blank t open square brackets negative fraction numerator 1 over denominator 2 end fraction g t close square brackets stack k with hat on top
    equals negative fraction numerator 1 over denominator 2 end fraction m g v subscript 0 end subscript t to the power of 2 end exponent c o s theta stack k with hat on top

    A small particle of mass m blankis projected at an angle theta with the x-axis with an initial velocity v subscript 0 end subscript in the x-y plane as shown in the figure. A t blank a blank t i m e blank t less than fraction numerator v subscript 0 end subscript sin invisible function application theta over denominator g end fraction comma the angular momentum of the particle is

    physics-General
    L equals m open parentheses r cross times v close parentheses
    L equals m open square brackets v subscript 0 end subscript c o s theta blank t stack stack i with dot on top with hat on top plus open parentheses v subscript 0 end subscript s i n theta blank t minus fraction numerator 1 over denominator 2 end fraction g t to the power of 2 end exponent close parentheses stack stack j with dot on top with hat on top close square brackets
    cross times open square brackets v subscript 0 end subscript c o s theta blank stack stack i with dot on top with hat on top plus open parentheses v subscript 0 end subscript s i n theta minus g t close parentheses stack stack j with dot on top with hat on top close square brackets
    equals m v subscript 0 end subscript c o s theta blank t open square brackets negative fraction numerator 1 over denominator 2 end fraction g t close square brackets stack k with hat on top
    equals negative fraction numerator 1 over denominator 2 end fraction m g v subscript 0 end subscript t to the power of 2 end exponent c o s theta stack k with hat on top