6. Rotational Motion Q

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ROTATIONAL MOTION CET WORKSHEETS

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The moment of inertia of a body comes into play a) in the rotational motion b) in a motion along elliptical or parabolic path c) in a motion along curved path d) in a motion along linear path In rotational motion, the moment of inertia I is analogous to the a) torque ‘ τ ’ in rotational motion b) mass in translational motion c) angular velocity in rational motion d) force in translation motion Consider a uniform circular ring of mass M and radius, R the moment of inertia i. about an axis a) MR 2 passing through 2 its centre and perpendicular to its plane ii. about diameter b) 2MR 2 iii. about a tangent in c) MR 2 its plane (parallel to plane) iv. about tangent and d) 3 MR 2 perpendicular to 2 its plane a) i – A, ii – B, iii – C, iv – D b) i – D, ii – C, iii – B, iv – A c) i – C, ii – A, iii – D, iv – B d) i – C, ii – A, iii – B, iv - D Consider a solid sphere of mass M and radius R, the moment of inertia about an axis passing through its centre and i. perpendicular to its plane is ii. about diameter is iii. about tangent in the plane (parallel) iv. about tangent perpendicular to its plane is a) i – B, ii – B, iii – A, iv – A b) i – A, ii – B, iii – B, iv – A c) i – B, ii – A, iii – B, iv – A d) i – B, ii – A, iii – A, iv - B A ring and a disc have same mass and same radius. Ratio of moment of inertia of the ring about a tangent in its plane to that of disc about its diameter is a) 9 : 1 b) 6 : 1 c) 1 : 6 d) 9 : 1

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Ratio of MI of a ring about the tangent normal to its plane to the MI of the ring about a tangent in the plane of the ring is a) 3 : 4 b) 3 : 2 c) 2 : 3 d) 4 : 3 MI of a solid cylinder of radius R and height about its own axis is (density is ρ ) 1 π R2 ρ h a) b) π R 4 ρ h 4 1 3 1 π R ρh π R4ρh c) d) 3 2 Moment of inertia of earth about its axis of rotation is (R is radius and ρ is density) 15 5 8 πR ρ π R5 ρ a) b) 8 15 2 3 4 5 πR ρ πR ρ c) d) 3 9 What is the M.I. of a thin uniform ring of mass 2kg and diameter 1 m rotating about the axis passing through ins centre and perpendicular to its plane of the ring a) 1kg m 2 b) 10 kg .m 2 c) 0.5 kg .m 2 d) 5 kg .m 2 Which of the following bodies having same mass and same radius has largest moment of inertia ? a) ring about its axis perpendicular to its plane b) disc about its axis perpendicular to its plane c) solid sphere d) bar magnet For a uniform rectangular plate of mass M, length L and breadth b, rotating about transverse axis through its centre is a)

(

M L2 + b 2

)

b)

(

M L2 + b 2

12

c) 12.

(

M L2 + b 2

)

8

)

d)

(

M L2 + b 2

)

10 6 Two cylinders are of equal mass and has same M.I. about axis. But one of them is solid and other hollow. What will be ratio of their diameter a) b) 1 : 2 2 :1

c)

1: 2

d)

2:1 Rotational Motion

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The masses are fixed on a massless rod as shown in figure. The moment of inertia about the given axis is a) 2 kgm 2 b)

1.04 kgm 2

c)

0.64 kgm 2

d)

0.32 kgm 2

Radius of gyration of a body about an axis gives the idea of a) volume of body b) area of given body c) distribution of mass of body d) shape and size of body Which of the following has the highest radius of gyration when each of them has the same diameter and mass a) a ring about an axis perpendicular to its plane b) a disc about an axis perpendicular to its plane c) a spherical shell about one of its diameter d) a solid sphere about one of its diameter The radius of gyration of a rod of mass 100 g and length 100 cm about an axis passing through its centre and perpendicular to its length will be a) 0.289 m b) 0.58 m c) 0.91 m d) 0.78 m The radius of gyration of disc of mass 100 gm and radius 5 cm about an axis passing through its centre of gravity and perpendicular to the plane is a) 1.5 cm b) 3.54 m c) 2.91 cm d) 3.54 cm The ratio of radii of gyration of a circular ring and a disc of the same radius about the axis passing through their centres and perpendicular to their planes is a) 1: 2 b) 2 :1 c) 2 : 1 d) 1 : 4 K.E. is a) scalar quantity b) vector quantity c) dimensionless quantity d) unitless quantity

Rotational Motion

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Which of the following is correct about ( K .E.)rot 1 mK 2ω 2 2 1 = Lω 2

i.

( K .E.)rot =

ii.

( K .E.)rot

iii.

( K .E.)rot =

iv.

( K .E.)rot

v.

( K .E.)rot =

1 L2 2 I = Lπ n

πL

T a) only (i) and (ii) are correct b) (i), (ii), (iv) and (v) are correct c) (i), (ii) and (iii) are correct d) (i), (ii), (iii), (iv) and (v) are correct A hollow cylinder and solid cylinder having same mass and same diameter are released from rest from the top of inclined plane, which will reach tot eh bottom first a) the solid cylinder b) he hollow cylinder c) both will reach to the bottom together d) nothing can be perdicted A flywheel of mass 60 kg and radius of gyration 40 cm is revolving at 30 rev/min, its K.E. is b) 14.4 π 2 J a) 3.6 π 2 J

d) 7.2 π 2 J c) 4.8π 2 J A flywheel has mass of 5 kg and radius of gyration 0.2 m. If it makes 300 r. p. m. then its rotational K.E. will be a) 10 J b) 98.7 J c) 19 J d) 987 J A circular disc of mass of 3 kg and diameter 0.4 m is rotating about an axis passing through its centre and perpendicular to its plane, if it makes 120 r. p. m. . Its rotational K.E. is b) 0.12π 2 a) 0.96π 2 c) 0.24π 2 d) 0.48π 2 A hollow spherical ball of mass 0.2 kg and radius 5 cm is rotating about an axis passing through its centre and making 300 r.p.m., its K.E. of rotation will be 5 × 10−3 π 2 J b) 100π 2 J a) 3 3 × 10−3 π 2 J c) 50 × 10−3 π 2 J d) 5

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Energy of 484 J is spent in increasing the speed of a flywheel from 60 r.p.m. to 360 r.p.m.. The MI of the wheel is b) 4.8 kg . m 2 a) 0.2 kg. m 2 c) 0.7 kg. m 2 d) 72 kg . m 2 A wheel of mass 2kg, having practically all the mass concentrated along the circumference of circle of radius 20 cm is rotating on its axis with angular velocity of 100 rad/s. The rotational K.E. of wheel is a) 40 J b) 400 J c) 800 J d) 80 J The M.I. of a wheel about its axis is 3kgm 2 its K.E. is 600 J . Its period of rotation will be a) 3.18 sec b) 0.314 sec c) 20 sec d) 0.05 sec A metal ring of mass 2.1 kg and 10 cm radius is revolving about its axis, making 100 / π revolution per sec. If this ring is dropped in a viscous liquid the heat generated in the liquid is (J = 4.2 joule/cal) a) 10 cal b) 1000 cal c) 100 cal d) 210 cal A body is kept rotating at constant speed by applying a couple of 10−3 Nm. When couple is removed it comes to rest in 20 revolutions. What was the initial kinetic energy of a body a) 2π × 10−2 J b) 6π × 10−2 J − 2 c) 4π × 10 J d) 8π × 10−2 J What is the torpue of force
 F = 2i − 3 j + 4k m acting at the point

( ) r = ( 3i + 2 j + 3k ) m

32.

a)

6i − 6 j + 12k

b)

−6i − 6 j − 12k

c)

17i − 6 j − 13k

about origin

d) −17i + 6 j + 13k A constant torque is applied on a circular wheel which changes its angular momentum form 1 J. sec to 4 J. sec in 4 sec. The torque will be 4 Nm b) 1 N m a) 3 3 8 Nm Nm c) d) 4 3

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A wheel has moment of inertia of 2kgm 2 and initial angular velocity of 40 rad/sec. When a constant torque of 8 N.m acts on it. The time in which the wheel is accelerated to 60 rad/sec is a) 20 sec b) 30 sec c) 10 sec d) 5 sec A ring starts from rest and requires an angular speed of 10 rad/s in 2 s. The mass of the ring is 500 g and its radius is 20 cm. The torque on the ring is a) 0.6 N.m b) 0.2 N.m c) 0.1 N.m d) 0.91 N.m Torque of magnitude 5 N.m is required for setting a thin uniform circular disc into rotation about the axis passing through the centre of the disc and normal to its plane. If the disc has to rotate above its diameter with the same angular acceleration then the required torque will be a) 5 N.m b) 2.5 N.m c) 10 N.m d) 15 N.m Torque of magnitude 20 N.m is required for spinning a uniform hollow sphere about its diameter with certain angular acceleration α . The torque required for spinning a uniform solid sphere of same mass and same radius as the hollow sphere about its diameter with angular acceleration 2α will be a) 30 N.m b) 40 N.m c) 24 N.m d) 10 N.m A torque of magnitude 400 N.m action on a body of mass 40 kg produces an angular acceleration of 20 rad / s 2 . The radius of gyration is a) 7 m b) 0.707 m c) 2 m d) 1.41 m Two rings of equal masses and thickness but of different material are acted upon by the same torque about an axis passing through its centre and perpendicular to its plane of the ring. If the densities of the materials are in the ratio of 4:1. The ratio of their angular acceleration will be a) 4 : 1 b) 16 : 1 c) 1 : 16 d) 1 : 4 A thin uniform rod mass M and length 2L is made to rotate about an axis passing through its centre and perpendicular to length. If its angular velocity changes from 0 to ω in time t, the torque acting on it is a)

ML2ω

b)

c)

ML2ω 3t

d)

ML2ω 2t ML2ω 4t Rotational Motion

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A flywheel rotating at 10 rev/sec is brought to rest by a constant torque in 15 sec, in coming to stop the flywheel would make a) 150 rev b) 75 rev c) 300 rev d) 600 rev A diesel engine develops a power of 90 KW at 2000 r.p.m. The average torque it can exert at the angular speed is b) 5 × 101 N .m a) 4 × 103 N .m c)

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4.3 × 102 N .m

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4 × 105 N .m

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A wheel of M.I. 5 ×10−3 kgm 2 is making 20 rev/s. It is brought to rest in 10 sec by applying a constant torque. If the radius of the wheel is 10 cm, the torque applied on the wheel is a) 2π × 10−3 N .m b) 50 π N .m d) 20π × 10−3 N .m c) 100 π N .m When a ceiling fan is switched on, it makes 10 rotations in the first 3 seconds. How many rotations will it make in the next 3 seconds (assume uniform angular acceleration) a) 10 b) 30 c) 20 d) 50 A wheel of mass 40 kg and radius of gyration 0.5 m comes to rest from a speed of 1800 rpm in 30 sec. Assuming that retarding torque is uniform, the value of retarding torque in Nm is b) 30π a) 10π c) 20π d) 40π Starting from rest a fan takes 6 second to attain the maximum speed of 30 rpm. Assuming constant acceleration, the time taken by fan in attaining half the maximum speed is a) 20 sec b) 2.5 sec c) 10 sec d) 3 sec The M.I. of a uniform circular disc about a diameter is I. The M.I. about an axis perpendicular to its plane and passing through a point on its rim will be a) I b) 6I c) 3I d) 2I The moment of inertia of a rod (length L, mass M) about an axis perpendicular to the length of the rod and passing through a point equidistant from its mid point and one end is 7 2 2 ML2 ML a) b) 48 5 12 2 48 2 ML ML d) c) 16 7

Rotational Motion

The moment of inertia of a uniform thin rod of length L and mass M about an axis passing L from one of through a point at a distance of 3 its ends and perpendicular to the rod is 7 1 ML2 ML2 b) a) 48 12 1 1 2 ML2 ML d) c) 9 3 Two sphere each of mass M and radius R/2 are connected with a massless rod of length 2R as shown in fig. What will be the moment of inertia of the system about an axis passing through the centre of one of the sphere and perpendicular to the rod

2 5 MR 2 MR 2 b) 5 4 21 MR 2 c) d) 5MR 2 5 Three rings each of mass M and radius R are arranged as shown in fig. The moment of inertia of the system about yy′ will be

a)

50.

a)

MR 2

b)

c)

2MR 2

d)

3MR 2 7 MR 2 2

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In a rectangle ABCD, BC = 2AB. Then M.I. along which axis is minimum

a) BD b) HF c) EG d) AB Four spheres of diameter 2a, mass M each are placed with their centres on the four corners of a square of side b. The M.I. of the system about one side of the square taken as its axis is 5 M a 2 + b2 a) 2 b) c)

2

2

(

Four solid spheres each of mass M and radius R are arranged as shown in fig. The M.I. of the system about yy

a)

55.

2

2

a)

56.

2 Ma

c)

2 Mb 2

5MR 2

4MR 2

d)

b) d)

1 Ma 2 2 1 Mb 2 2

c) 57.

ρ L2 8π 2 3 ρ L2 2 π2

b)

3ρ ML2 8π 2

2 ρ L2 3 π2 If h is the distance between two parallel axis and K c is the radius of gyration passing through centre of mass of a body. Then radius of gyration through any point is

a) a)

b)

)

c)

2

18 MR 2 5

5 MR 2 18 The thin wire of length ‘L’ and uniform mass density ρ is bent into a circular loop with centre at O as shown fig, the M.I. of the loop about the axis xx′ is

c)

2 M 4a 2 + 5b 2 5 Four point masses are fastened to the frame of negligible masses as shown. M and M along Z axis and M and M along Y-axis. What is the M.I. of the system when rotated about x-axis

d)

53.

( ) 2M ( a + b ) 2 M (a + b ) 5

54.

K C2 − h 2 KC2 + h 2

d)

b)

h 2 − K C2

d)

KC − h

The radius of gyration of a body about an axis at a distance of 12 cm from its centre of mass is 13 cm. Find its radius of gyration about a parallel axis passing through its centre of mass a) 5 cm b) 25 cm c) 12 cm d) 7 cm

Rotational Motion

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Four particles of masses 4 kg, 2 kg, 3kg and 5 kg are respectively located at the four corners A, B, C and D of a square of side 1 m, as shown in fig. The M.I. of the system about an axis passing through the point of interection of the diagonals and perpendicular to the plane of the square.

a) 59.

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b)

8kg − m 2

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d) 14 kg − m 2 c) 7 kg − m2 In the above question the M.I. of the system about the diagonal BD is a) 3.5 kg .m 2 b) 8kg − m 2

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d) 14 kg − m 2 c) 7 kg − m2 The M.I. of a cylinder of length 1.5 m, radius 0.05m and density 8 × 103 kg m −3 about the axis of the cylinder is a) 5 × 102 kg − m2 b) 9.1 × 103 kg − m2 d) 105.97 kg − m 2 c) 15 × 103 kg − m 2 The moment of inertia of two spheres of equal masses about their respective diameters are same. One of them is solid and other is hollow. The ratio of diameters of solid sphere to that of the hollow sphere is 3: 5 b) 5 : 3 a) 5: 3

d)

3:5

A solid sphere of mass 2 kg rolls up a 300 incline with an initial speed of 10 m/s. The maximum height reached by the sphere is

( g = 10 m / s ) 2

a) c)

3.5 m 7m

Rotational Motion

b) d)

10.5 m 14.0 m

The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to plate is

( I1 and I 2 are M.I. of square plate about an axis through centre O and perpendicular to plate and l is its length) b) I1 − I 2 a) I1 + I 2

c) 7 kg − m2 d) 14 kg − m 2 In the above question the M.I. of the system about the side AB is a) 3.5 kg .m 2 b) 8kg − m 2

c) 63.

3.5 kg .m 2

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c) I1I 2 d) I O = I1 + ml 2 Linear momentum is a) scalar quantity b) vector quantity c) dimensionless quantity d) tensor quantity The unit of L/p is a) meter b) sec c) m/s d) newton If L is angular momentum of rotating body and E is its rotational K.E. then 2E E2 b) L = a) L = ω ω E E c) L = 2 d) L = 2ω ω A body suddenly comes and suddenly sits on circular rotating table. What will remain cnserved a) angular velocity b) angular momentum c) angular acceleration d) linear momentum A person standing on the rotating platform with his hands lowered, suddenly oustretches his arms. The angular momentum of person a) increases b) decreases c) become zero d) remains constant A particle of mass M is moving in a horizontal circle of radius R with uniform speed v. When it moves from one point to diametrically opposite point its a) K.E changes by mv 2 b) momentum changes by 2 mv c) momentum does not changes d) K.E. chages by mv 2 / 4

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A dancer on ice spins faster (about the axis), when she folds her arms. It is due to a) increase in energy and increase in angular momentum b) decrease in energy and increase in angular moment c) constant angular momentum and increace in kinetic energy d) constant angular momentum and decease in kinectic enregy A body of mass M and radius of gyration K, rotates with angular velocity ω . What will be the angular momentum of the body a) Iω2 b) MK 2ω c) MKω d) I 2ω2 A merry-go-round is at rest, pivoted on a frictionless axis. A child of mass M runs along a path tangential to the rim with speed v and jumps on the merry-go-round. If R is the radius of the merry-go-round, L is the angular velocity of merry-go-round, I be the moment of inertia of merry-go-round. Angular velocity of merry-goround and child is MvR MvR a) b) I MR 2 + I I MR 2 + I c) d) MvR MvR A pan containing a layer of uniform thickness of ice is placed on a circular turntable with its centre coinciding with the centre of constant angular velocity about a vertical axis passing through its centre and driving torque is withdrawn. There is no friction between the table and pivot. As ice melts a) the angular velocity decreases b) the angular velocity increases c) the angular velocity remains unchanged d) the moment of inertia remains unchanged A horizontal disc rotating freely about vertical axis makes 90 rpm. A small piece of wax of mass ‘m’ gram falls vertically on the disc and sticks to it at a distance of r cm from the axis. If number of revolutions per mimute reduce to 60 rpm the moment of inertia of the disc is b) 2 mr 2 a) mr 2 c)

3 mr 2 2

d)

3mr 2

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A thin uniform circular disc of mass M and radius R in a horizontal plane about an axis passing through its centre and angular velocity ω . Another disc of same dimensions but M of mass is placed on the first disc co-axially. 4 The angular velocity of the system is now 4 ω b) a) 2ω 5 ω d) c) 2ω 3 If the Earth is considered a uniform sphere of mass 6 × 1024 kg and radius 6.4 ×106 m spining on its axis at the rate of one turn per day. What is the angular momentum associated with this spins a) 7.145 × 1033 kg m 2 / sec b)

7.145 × 1030 kg m 2 / sec

c)

71.45 × 1037 kg m 2 / sec

d)

714 × 1020 kg m 2 / sec

A disc of M.I. I1 is rotating freely with angular speed ω1 . When another non rotating disc with M.I. I 2 is dropped on it. The two now rotates as a unit, final angular speed is b) ω1 ( I1 + I 2 ) / I1 a) ( I1 + I 2 ) ω1

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d) I1ω1 / ( I1 + I 2 ) c) ( I1 − I 2 ) ω1 A thin circular ring of mass ‘M’ and radius ‘R’ rotating about its axis with a constant angular speed ω . Two blocks each of mass ‘m’ are attached gently to opposite ends of a diameter of the ring. The angular speed of the ring will be 2M ω ( M − 2m ) ω a) b) ( M + 2m ) M + 2m Mω ( M + 2m ) ω c) d) ( M + 2m ) M If radius of Earth suddenly constracts to 1 / nth of its present value without any change in its mass, the duration of one day will become approximately 24 hr a) b) 24 n hr n 24 hr d) c) 24 × n 2 hr n2

Rotational Motion

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If spherical Earth were transformed to hollow sphere of same mass and radius without application of external torque then angular velocity of the rotating earth would be a) 25 % of its initial value b) 40 % of its initial value c) 60% of its initial value d) 80 % of its intial value A wheel is rotating with an angular speed of 750 r.p.m. on a shaft. Another identical wheel is suddenly coupled on the same shaft. The frequency of the system will be (assuming M.I. of shaft to be negligible) a) 375 r.p.m. b) 750 r.p.m c) 500 r.p.m d) 163 r.p.m A ballet dancer spins about a vertical axis at 180 r.p.m. with her arms stretched. When she folds her arms MI about the same axis decreases by 40%. Her new speed will be a) 180 r.p.m. b) 150 r.p.m c) 72 r.p.m d) 300 r.p.m A torque of 50 Nm acts one a rotating body for 5 sec. Its angular momentum a) increases by 250 kg m 2 / s b)

increases by 10 kg m 2 / 2

c)

increases by 55 kg m 2 / s

d) decreases by 250 kg m 2 / s A loop of mass M and radius R is rolling on a smooth horizontal surface with speed v its total K.E. is 1 2 mv a) b) mv 2 2 3 2 1 mv mR 2ω2 c) d) 2 2 The speed of a homogeneous solid sphere of mass m and M.I. I rolling down an inclined plane of vertical height ‘h’ from the rest without sliding is 6 10 b) a) gh gh 5 7 4 gh c) d) gh 3 The speed of a homogeneous disc of mass m and M.I. I rolling down an inclined plane vertical height ‘h’ from the rest without sliding is 4 10 b) a) gh gh 3 7

c)

gh

Rotational Motion

d)

6 gh 5

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An inclined plane makes an angle 450 with the horizontal. A solid sphere rolling down this inclined plane from rest without slipping has a linear acceleration equal to g 5 2 b) a) g 14 14 g d) 5 2 g c) 2 A body is rolling down an inclined plane velocity of body is independent of a) angle of inclination b) height through which body descends c) both ‘a’ and ‘c’ d) neither ‘a’ nor ‘b’ If a body is rolling down an inclined plane. Its acceleration is ‘a’ and inclination is θ , if θ increases then ‘a’ will be a) greater b) lesser c) same d) can’t say Two identical cylinders are released from the top of two identical inclined planes. If one rolls without slipping and other slips without rolling then, a) rolling cylinder reaches the bottom first with greater speed b) sliding cylinder reaches the bottom first with greater speed c) both reaches the bottom simultaneously and with the same speed d) both reaches the bottom simultaneously but with the different speeds A spherical shell first rolls down an inclined plane and then slides down. What will be the ratio of the speeds at the bottom in the two cases 3 5 b) a) 5 3 1 3 d) c) 3 5 A body of radius R and mass M is rolling horizontally without slipping with speed v. It then rolls up a hill to a maximum height

3v 2 , 4g

what might the body be (neglect friction) a) disc b) hollow cylinder c) solid cylinder d) both ‘a’ and ‘c’

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A solid cylinder of mass M and radius R rolls down on inclined plane without slipping. The speed of its centre of mass when it reaches the bottom is 4 3 b) a) gh gh 3 4 books Not given 2gh d) c) d opt. Which of the following statements is/are correct for a body of same mass and same radius i. ( I )ring > ( I )h.shpere > ( I )disc > ( I )s.sphere ii. ( v ) s.sphere > ( v ) disc > ( v )h.sphere > ( v ) ring iii. ( t )s.sphere < ( t )disc < ( t )h.sphere < ( t )ring iv. ( t )s.sphere = ( t )disc = ( t )h.sphere = ( I )ring

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a) i, ii and iii are correct b) i, ii and iv are correct c) ii and iv are correct d) i and iii are correct When a sphere of moment of inertia (I) moves down on inclined plane, the percentage of energy which is rotational is approximately a) 72 % b) 28 % c) 18 % d) 100 % When a ring rolls down an inclined plane the percentage of energy which translational K.E. and rotational K.E. a) 50 % and 50 % b) 75% and 25 % c) 25% and 75 % d) 30 % and 70 % A solid sphere rolls down on inclined plane the percentage of total energy which is rotational K.E. is a) 28 % b) 100 % c) 72 % d) 75% A thin hoop of diameter 0.5 m and mass 2 kg rolls down on inclined plane from rest. If its linear speed on reaching the foot of the plane is 2m/s its rotational K.E. at the instant is a) 2 J b) 4 J c) 3 J d) 6 J A body rolls down on inclined plane. If its kinetic energy of rotational motion is 40% of its kinetic energy of translational then the body is a) disc b) ring c) hollow sphere d) solid sphere

Rotational Motion