General
Easy
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

Physics-General

  1. m u to the power of 2 end exponent sin invisible function application theta    
  2. fraction numerator 1 over denominator 2 end fraction m u to the power of 2 end exponent cos invisible function application 2 theta    
  3. m u to the power of 2 end exponent cos invisible function application theta    
  4. fraction numerator 1 over denominator 2 end fraction blank m u to the power of 2 end exponent sin invisible function application 2 theta    

    Answer:The correct answer is: fraction numerator 1 over denominator 2 end fraction blank m u to the power of 2 end exponent sin invisible function application 2 thetaTime 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

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    Related Questions to study

    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-

    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
    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 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 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
    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 piece of wire is bent in the shape of a parabola y=kx2 (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

    macosθ=mgcos(90-θ) ag=tanθag=dydx ddxkx2=agx=a2gk    

    A piece of wire is bent in the shape of a parabola y=kx2 (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
    macosθ=mgcos(90-θ) ag=tanθag=dydx ddxkx2=agx=a2gk    
    General
    physics-

    A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

    Velocity of the bob at the point A v=5gL(i) v22=v2-2ghii h=L(1-cosθ)(iii) Solving Eqs.i, iiand iii, we get cosθ=-78 or   θ=cos-1-78=151°    

    A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

    physics-General
    Velocity of the bob at the point A v=5gL(i) v22=v2-2ghii h=L(1-cosθ)(iii) Solving Eqs.i, iiand iii, we get cosθ=-78 or   θ=cos-1-78=151°    
    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
    maths-

    The shortest distance between the two straight linex4/32=y+6/53=z3/24 and 5y+68=2z39=3x45 is  

    The shortest distance between the two straight linex4/32=y+6/53=z3/24 and 5y+68=2z39=3x45 is  

    maths-General
    General
    maths-

    The value of 0100{x}dx (where {x} is the fractional part of x) is 

       

    The value of 0100{x}dx (where {x} is the fractional part of x) is 

    maths-General
       
    General
    maths-

    The value of 01|sin 2πx|dx is equal to 

    Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.    

    The value of 01|sin 2πx|dx is equal to 

    maths-General
    Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0 but normal to the plane will be perpendicular to x-axis. Hence (c) is the correct answer.    
    General
    maths-

    Let f:RR,f(x)=|x[x]|,[x]|x[x+1]|,[x] is odd 1 is even where [.]  denotes greatest integer function, then 24f(x)dx is equal to

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

    Let f:RR,f(x)=|x[x]|,[x]|x[x+1]|,[x] is odd 1 is even where [.]  denotes greatest integer function, then 24f(x)dx 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.