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 Multiple Choice QuestionsMultiple Choice Questions

1.

A hoop of radius r and mass m rotating with an angular velocity ω0
is placed on a rough horizontal surface.The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

  • 0/4

  • 0/3

  • 0/2

  • 0


C.

0/2

From conservation of angular momentum about any fixed point on the surfacemr squared straight omega subscript 0 space equals space 2 mr squared straight omega
therefore space straight omega space equals space straight omega subscript 0 over 2
therefore space straight V subscript CM space equals fraction numerator straight omega subscript 0 straight r over denominator 2 end fraction space

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2.

A cylindrical tube, open at both ends, has a fundamental frequency, f, in the air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now

  • m1r1:m2r2

  • m1 :m2

  • r1 :r2

  • 1:1


C.

r1 :r2

As their period of revolution is same, so its angular speed is also same. Centripetal acceleration is circular path,
a= ω2r
Thus, 
straight a subscript 1 over straight a subscript 2 space equals space fraction numerator straight omega squared straight r subscript 1 over denominator straight omega squared straight r subscript 2 end fraction space equals space straight r subscript 1 over straight r subscript 2

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3.

From a solid sphere of mass M and radius R, a cube of the maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is

  • fraction numerator MR squared over denominator 32 square root of 2 straight pi end root end fraction
  • fraction numerator MR squared over denominator 16 square root of 2 straight pi end root end fraction
  • fraction numerator 4 MR squared over denominator 9 square root of 3 straight pi end root end fraction
  • fraction numerator 4 MR squared over denominator 3 square root of 3 straight pi end root end fraction

C.

fraction numerator 4 MR squared over denominator 9 square root of 3 straight pi end root end fraction

Consider the cross-sectional view of a diametric plane as given the figure.
Using geometry of the cube


PQ space equals space 2 straight R space equals space left parenthesis square root of 3 right parenthesis space straight a space or space space straight a space equals space fraction numerator 2 straight R over denominator square root of 3 end fraction
Volume density of the solid sphere

straight rho space equals space fraction numerator straight M over denominator begin display style 4 over 3 end style πR cubed end fraction space equals space fraction numerator 3 over denominator 4 straight pi end fraction open parentheses straight M over straight R cubed close parentheses
mass space of space cube space left parenthesis straight m right parenthesis space space equals space left parenthesis straight rho right parenthesis left parenthesis straight a cubed right parenthesis
space equals space open parentheses fraction numerator 3 over denominator 4 straight pi end fraction straight x space straight M over straight R cubed close parentheses open square brackets fraction numerator 2 straight R over denominator square root of 3 end fraction close square brackets cubed
space equals space fraction numerator 3 straight M over denominator 4 πR cubed end fraction space straight x space fraction numerator 8 straight R cubed over denominator 3 square root of 3 end fraction space equals fraction numerator 2 straight M over denominator square root of 3 straight pi end fraction
Moment of inertia of the cube about the given axis is 
straight I subscript straight y space equals space ma squared over 12 space left parenthesis straight a squared plus straight a squared right parenthesis space space equals ma squared over 6
rightwards double arrow space straight I subscript straight Y space equals space ma squared over 6 space equals space fraction numerator 2 straight M over denominator square root of 3 straight pi end root end fraction space straight x 1 over 6 space straight x space fraction numerator 4 straight R squared over denominator 3 end fraction space equals space fraction numerator 4 MR squared over denominator 9 square root of 3 straight pi end root end fraction

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4.

Two identical charged spheres suspended from a common point by two massless strings of length l are initially a distance d(d < < l) apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate.As a result the charges approach each other with a velocity v. Then as a function of distance x between them

  • v ∝ x-1

  • v ∝ x1/2

  • v ∝ x

  • v ∝ x-1/2


D.

v ∝ x-1/2



237 Views

5.

From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its centre and perpendicular to one of its faces is :

  • h2/4R

  • 3h/4

  • 5h/8

  • 3h2/8R


B.

3h/4

We know that centre of mass of uniform solid cone of height h is at height h/4 from base therefore,




h-zo = h/4
zo = h-h/4 = 3h/4

740 Views

6.

This question has Statement I and Statement II. Of the four choices given after the Statements, choose the
one that best describes the two Statements.
Statement – I: A point particle of mass m moving with speed v collides with stationary point particle of mass M. If the maximum energy loss possible is given as f open parentheses 1 half mv squared close parentheses comma space then space straight f space equals space open parentheses fraction numerator straight m over denominator straight M plus straight m end fraction close parentheses
Statement – II : Maximum energy loss occurs when the particles get stuck together as a result of the collision.

  • Statement – I is true, Statement – II is true, Statement – II is a correct explanation of Statement – I.

  • Statement – I is true, Statement – II is true, Statement – II is not a correct explanation of Statement – I.

  • Statement – I is true, Statement – II is false.

  • Statement – I is false, Statement – II is true 


D.

Statement – I is false, Statement – II is true 

Before collision, the mass is m and after collision, the mass is m+M
therefore, Maximum energy loss
fraction numerator straight p squared over denominator 2 straight m end fraction minus fraction numerator straight p squared over denominator 2 left parenthesis straight m plus straight M right parenthesis end fraction
space equals space fraction numerator straight p squared over denominator 2 straight m end fraction open square brackets fraction numerator begin display style straight m end style over denominator straight m plus straight M end fraction close square brackets space space space space
space space space open square brackets because KE space equals space fraction numerator straight p squared over denominator 2 straight m end fraction close square brackets
equals space 1 half mv squared open curly brackets fraction numerator straight m over denominator straight m plus straight M end fraction close curly brackets
open square brackets straight f space equals space fraction numerator straight m over denominator straight m plus straight M end fraction close square brackets

398 Views

7.

A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed ω rad/s about the vertical. About the point of suspension

  • angular momentum is conserved

  • angular momentum changes in magnitude but not in the direction

  • angular momentum changes in direction but not in magnitude

  • angular momentum changes both in direction and magnitude


C.

angular momentum changes in direction but not in magnitude

542 Views

8.

A thin horizontal circular disc is rotating about a vertical axis passing through its centre. An insect is at rest at a point near the rim of the disc. The insect now moves along a diameter of the disc to reach its other end. During the journey of the insect, the angular speed of the disc 

  • remains unchanged

  • continuously decreases

  • continuously increases 

  • first increases and then decreases


D.

first increases and then decreases


From angular momentum conservation about the vertical axis passing through centre. When the insect is coming from circumference to center. Moment of inertia first decreases then increase. So angular velocity increase than decrease. 
633 Views

9.

A metallic rod of length ‘l’ is tied to a string of length 2l and made to rotate with angular speed ω on a horizontal table with one end of the string fixed. If there is a vertical magnetic field ‘B’ in the region, the e.m.f. induced across the ends of the rod is

  • fraction numerator 2 Bωl cubed over denominator 2 end fraction
  • fraction numerator 3 Bωl cubed over denominator 2 end fraction
  • fraction numerator 4 Bωl squared over denominator 2 end fraction
  • fraction numerator 5 Bωl squared over denominator 2 end fraction

D.

fraction numerator 5 Bωl squared over denominator 2 end fraction


de space equals space straight B space left parenthesis ωx right parenthesis. dx
straight e space equals space Bω integral subscript 2 straight L end subscript superscript 3 straight L end superscript xdx
space equals fraction numerator 5 BωL squared over denominator 2 end fraction
1106 Views

10.

A diatomic molecule is made of two masses m1 and m2 which are separated by a distance r. If we calculate its rotational energy by applying Bohr's rule of angular momentum quantization, its energy will be given by (n is an integer)

  • fraction numerator left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis squared straight n squared straight h squared over denominator 2 straight m subscript 1 superscript 2 straight m subscript 2 superscript 2 straight r squared end fraction
  • fraction numerator straight n squared straight h squared over denominator 2 left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis straight r squared end fraction
  • fraction numerator 2 straight n squared straight h squared over denominator left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis straight r squared end fraction
  • fraction numerator left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis straight n squared straight h squared over denominator 2 straight m subscript 1 straight m subscript 2 straight r squared end fraction

D.

fraction numerator left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis straight n squared straight h squared over denominator 2 straight m subscript 1 straight m subscript 2 straight r squared end fraction

Rotational kinetic energy of the two body system rotating about their centre of mass is
RKE space equals space 1 half μω squared straight r squared
where comma space straight mu space equals space fraction numerator straight m subscript 1 straight m subscript 2 over denominator straight m subscript 1 plus straight m subscript 2 end fraction equals space reduced space mass
and space angular space momentum comma space straight L space equals space μωr squared space equals space fraction numerator nh over denominator 2 straight pi end fraction
straight omega space equals fraction numerator nh over denominator 2 πμr squared end fraction
therefore space RKE space equals space 1 half μω squared straight r squared space equals space 1 half space straight mu. open parentheses fraction numerator nh over denominator 2 πμr squared end fraction close parentheses squared straight r squared
space equals fraction numerator straight n squared straight h squared over denominator 8 straight pi squared μr squared end fraction space equals space fraction numerator straight n squared straight ħ squared over denominator 2 μr squared end fraction space open parentheses where comma straight ħ space equals space fraction numerator straight h over denominator 2 straight pi end fraction close parentheses
fraction numerator left parenthesis straight m subscript 1 plus straight m subscript 2 right parenthesis straight n squared straight ħ squared over denominator 2 straight m subscript 1 straight m subscript 2 straight r squared end fraction

864 Views