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#[email protected] GURUKUL #Rotational Motion Level – 1 Question for Practice DPS-R1 1.

A ball slides without friction down an inclined plane of 5R height 2

2R

600 5R/2

4R 60

0

O

2.

3.

. It crosses the bottom of the inclined plane and moves along a circular loop of radius R. The force exerted by the ball at point A is (a) mg (b) 2.5 mg (c) 3.5 mg (d) 4.5 mg A ball weighing 15 g is tied to a string 10 cm long. Initially the ball is held in position such that the string is horizontal. The ball is now released. A nail N is situated vertically below 7. the support at a distance L. The minimum value of L such that the string will be wound round the nail is (a) 2 cm (b) 4 cm (c) 6 cm (d) 8 cm A block of mass M is placed on a horizontal frictionless surface AB and a body P of mass m is released on its 8. frictionless slope. C P M A

4.

5.

6.

Q

B

As P slides by a length L on this slope of inclination θ, the block Q would slide by a distance : m m L cos θ (b) L (a) M M+m M+m mL cos θ (d) (c) mL cos θ m+M A body of mass M and radius R is rolling horizontally without slipping with speed v. It then rolls up a hill to a maximum height h. If h = 5v2/6g, what is the M.I. of the body ? 2 1 (b) MR2 a) MR2 2 3 3 2 2 (d) MR2 (c) MR 4 5 A boy and a man carry a uniform rod of length l horizontally 1 in such a way that the boy gets th of the load. If the boy is 4 at one end of the rod, then the distance of the man from the other end is 3l l 2l l (b) (c) (d) (a) 3 3 4 4 A carpenter has constructed a toy as shown in the adjoining fig.

9.

If the density of the material of the sphere is 12 times that of cone, the position of the centre of mass of the toy is given by : (a) at a distance of 2R from O (b) at a distance of 3R from O (c) at a distance of 4R from O (d) at a distance of 5R from O A certain bicycle can go up a gentle incline with constant speed when the frictional force of ground pushing the rear wheel if F2 = 4 N. With what force F1 must the chain pull on the sprocket wheel if R1 = 5 cm and R2 = 30 cm ? (a) 4 N (b) 24 N 35 N c) 140 N d) 4 A heavy disc is thrown on a horizontal surface in such a way that it slides with a speed V0 initially without rolling. It will start rolling without slipping when its speed reduces to : V 2V0 (b) (a) 0 2 3 5V0 3V0 (c) d) 5 7 A heavy particle hanging from a string of length l is projected horizontally with speed

gl . The speed of the particle at the

point where the tension in the string equals the weight of the particle is (a)

(2gl)

(b)

(3gl)

(c)

(gl / 2)

(d)

(gl / 3)

10. A mass m is supported by a massless string wound around a uniform cylinder of mass m and radius R, Fig. With what acc. will the mass fall on release ? (a) 2g/3 b) g/2 c) g (d) 4g/3 11. A particle of mass m = 5 is moving with a uniform speed v = 3√2 in the XOY plane along the line y = x + 4. The magnitude of the angular momentum about origin is (a) zero b) 60 unit (c) 7.5 unit (d) 40 √2 12. A particle of mass m is rotating in a plane in circular path of radius r. Its angular momentum is L. The centripetal force acting on the particle is

(a)

L2 mr

b)

the normal of inertia of the same cylinder about an axis passing through its centre and normal to its length l then :

L2 m r2

(a) L = R (b) L = 3 R R (c) L = (d) L = 0 3 13. A particle originally at rest at the highest point of a smooth 19. A uniform solid right circular cone of base radius r is joined to a uniform solid hemisphere of radius r and of the same vertical circle is slightly displaced. It will leave the circle at a density, so as to have a common face. The centre of gravity vertical distance h below the highest point, such that of the composite solid lies on the common face. The height of (a) h = R b) h = 2R the cone is (c) h = R/2 (d) h = R/3 14. A rigid body rotates about a fixed axis with variable angular (b) r/ 6 a) 3r/ 2 velocity equal to α - βt at time t where α and β are constants. c) 4r (d) r 3 The angle through which it rotates before it comes to rest is : 20. Angular momentum and areal velocity of a body of mass m α 2 − β2 α2 (b) (a) are related as 2α 2β a) L = 2m × areal velocity 2 2 b) L = 2π × areal velocity α(α − β) α −β (c) (d) c) Areal velocity = 2mL 2 2β (d) 2m = 2 × areal velocity 15. A ring rolls down, starting from rest, down an inclined plane 21. Consider the rotation of a rod of mass m and length l from of length l and inclination θ. The velocity of the centre of position AB to AB/. mass of the ring at the mid-point of the inclined plane is 2

L d) . mr

L2 c) mr 3

gl sin θ

(a)

(b)

4 gl sin θ 7 16. A thick walled hollow sphere has outer radius R. It rolls down an inclined plane without slipping and its speed at the bottom is v. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom will be 5v/4. What is the radius of gyration of the sphere? R R (b) (a) 2 2 2gl sin θ

(c)

A

gl sin θ 2

(d)

B

B’

Which of the following is correct ? a) Weight of the rod is lowered by

l . 2

b) Loss of gravitational potential energy is (c) Angular velocity is

1 mgl. 2

3g . l

3R 3R (d) 4 4 ml2ω2 (d) Rotational kinetic energy is 17. A uniform circular disc of radius a is taken. A circular portion 3 of radius b has been removed from it as shown in the fig. 22. Fig. shows a thin metallic triangular sheet ABC. (c)

A a

l

B

b l

O

c

f the centre of hole is at a distance c from the centre of the disc, the distance x2 of the centre of the mass the remaining part from the initial centre of mass O is given by :

πb 2

C

The mass of the sheet is M. The moment of inertia of the sheet about side AC is :

− cb 2

Ml 2 Ml 2 Ml 2 Ml 2 (b) (c) (d) (a − b ) (a − b ) 18 12 6 4 23. From a given sample of uniform wire, two circular loops P 2 2 πc πa (c) 2 (d) and Q are made, P of radius r and Q of radius nr. If the M.I. (a − b 2 ) (c 2 − b 2 ) of Q about its axis is 4 times that of P about its axis 18. A uniform cylinder has a radius R and length L. If the (assuming wire diameter much smaller, than either radius), moment of inertia of this cylinder about an axis passing the value of n is : through its centre and normal to its circular face is equal to a)

2

2

(b)

2

2

(a)

(a) (4)2/3 b) (4)1/3 1/2 d) (4)1/4 c) (4) 24. If a disc slides from top to bottom of an inclined plane, it

Z

t2 takes time t1. If it rolls, it takes time t2. Now, 22 is t1 Y 2 1 (a) (b) x 2 3 2 3 The moment of inertia of this system about Z-axis is : (d) (c) 5 2 ML2 2ML2 4ML2 5ML2 (b) (c) (d) (a) 25. If a sphere is rolling, the ratio of the translational energy to 3 3 3 3 total kinetic energy is given by 32. Two discs, each of mass 1 kg and radius metre, are 2 a) 7:10 b) 2:5 placed parallel to each other and separated by a distance of (c) 10:7 d) 5:7 26. If the earth is treated as a sphere of radius R and mass M, its 2 metre. angular momentum about the axis of its diurnal rotation with Y period T is

2πMR 2 4πMR 2 MR 2T πMR 3 (b) (c) (d) 5T T 2π T 27. In fig. (a), a metre stick, half of which is wood and the other half steel is pivoted at the wooden end at O and a force F is applied to the steel end A. (a)

O Wood

Steel

A

F

Wood A/

F

(a ) O/ Steel

Y'

YY’ is an axis which is perpendicular to the line joining the centers of the two discs and is mid-way between the two discs. The moment of inertia of the system about YY’ is (b) 2 kg m2 (a) 1 kg m2 2 (d) 8 mg r12 (c) 4 kg m 33. Two thin discs each of mass M and radius r metre are attached as shown in fig., to form a rigid body.

b A In fig. (b) the stick is pivoted at the steel end at O’ and B the same force F is applied at the wooden end at A’. The angular acceleration: (a) in (a) is greater than in (b) The rotational inertia of this body about an axis (b) in (b) is greater than in (a) perpendicular to the plane of disc B and passing through its (c) is equal both in (a) and (b) centre is : (d) none of the above b) 3Mr2 a) 2Mr2 2 28. Moment of inertia of a uniform circular disc about a diameter d) 5Mr2 c) 4Mr is I. Its moment of inertia about an axis ⊥ to its plane and 34. Two uniform thin identical rods AB and CD each of mass M passing through a point on its rim will be and length L are joined so as to form a cross as shown. (a) 5I (b) 3I (c) 6I (d) 4I C F 29. The mean kinetic energy of a particle of mass m moving with constant force in any interval of time will be …. where ω1 A B and ω2 are the initial and final velocities. m m 2 (ω1 + ω1ω 2 + ω 22 ) (b) (ω1 + ω 2 ) 2 (a) E 6 2 D m m 2 2 The moment of inertia of the cross about a bisector line (ω1 + ω 2 ) (ω1 − ω 2 ) (d) (c) 3 6 EF is : 30. The rotational kinetic energy of a body is E and its moment ML2 ML2 ML2 ML2 of inertia is 1. The angular momentum is (b) (c) (d) (a) 6 4 12 3 (a) EI (b) 2 IE 35. When a ceiling fan is switched off, its angular velocity falls to half while it makes 36 rotations. How many more rotations (d) E/I. (c) 2 EI will it make before coming to rest ? (Assume uniform angular 31. Three thin rods each of length L and mass M are placed along retardation) X, Y and Z-axes in such a way that one end of each of the (a) 36 (b) 24 rods is at the origin. (c) 18 (d) 12

36. The moment of inertia of a uniform thin rod of length L and mass M about an axis passing through a point at a distance of L/3 from one of its ends and perpendicular to the rod is :

38.

39.

40.

41.

(a)

g L

(b)

1 2

g L

3 g g 7 ML2 ML2 (c) (d) 2 (b) 2 L L 48 1 42. A uniform rod of mass M and length L is pivoted at 2 ML ML2 one end such that it can rotate in a vertical plane. (d) (c) 9 3 There is negligible friction at the pivot. The free end Two thin discs each of mass M and radius r metre are of the rod is held vertically above the pivot and then attached as shown in fig., to form a rigid body. The released. The angular acceleration of the rod when it rotational inertia of this body about an axis makes an angle θ with the vertical is : perpendicular to the plane of disc B and passing g (a) g sin θ (b) sin θ through its centre is : l (b) 3Mr 2 (a) 2Mr 2 3 g (d) 5Mr 2 (c) 4Mr 2 sin θ (d) 6 gl sin θ (c) 2l A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord 43. A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force F is applied is wrapped around the rim of the disc and a steady normal to one of the faces at a point that is directly downward pull T is exerted on the cord. The angular 3a acceleration of the disc is : above the base. above the centre of face, at a height 4 MR T (b) (a) The minimum value of F for which the cube begins to T MR tilt about the edge is (assume that the cube does not MR 2T (d) (c) slide) : 2T MR 2mg mg A body is rolling without slipping on a horizontal (b) (a) 4 3 surface and its rotational kinetic energy is equal to the 3 mg translational kinetic energy. The body is : (d) mg (c) (a) disc (b) sphere 4 (c) cylinder (d) ring 44. A cubical block of mass M and edge a slides down a A particle is mass m is projected with a velocity v rough inclined plane of inclination θ with a uniform making an angle of 45° with the horizontal. The velocity. The torque of the normal force on the block magnitude of angular momentum of the projectile about its centre has a magnitude : about an axis of projection when the particle is at (a) zero (b) Mga maximum height h is : Mga sin θ (c) Mga sin θ (d) mv3 2 (a) zero (b) 45. A rigid spherical body is spinning around an axis 4 2g without any external torque. Due to change in mv 2 temperature, the volume increases by 1%. Its angular (c) (d) m 2gh 3 speed : 2g (a) will increase approximately by 1% A rigid body is made of three identical thin rods, each (b) will decrease approximately by 1% of length L fastened together in the form of letter H. (c) will decrease approximately by 0.67% The body is free to rotate about a horizontal axis that (d) will decrease approximately by 0.33% runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of H is horizontal. What is the angular (a)

37.

speed of the body when the plane of H is vertical ?

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A ball slides without friction down an inclined plane of 5R height 2

2R

600 5R/2

4R 60

0

O

2.

3.

. It crosses the bottom of the inclined plane and moves along a circular loop of radius R. The force exerted by the ball at point A is (a) mg (b) 2.5 mg (c) 3.5 mg (d) 4.5 mg A ball weighing 15 g is tied to a string 10 cm long. Initially the ball is held in position such that the string is horizontal. The ball is now released. A nail N is situated vertically below 7. the support at a distance L. The minimum value of L such that the string will be wound round the nail is (a) 2 cm (b) 4 cm (c) 6 cm (d) 8 cm A block of mass M is placed on a horizontal frictionless surface AB and a body P of mass m is released on its 8. frictionless slope. C P M A

4.

5.

6.

Q

B

As P slides by a length L on this slope of inclination θ, the block Q would slide by a distance : m m L cos θ (b) L (a) M M+m M+m mL cos θ (d) (c) mL cos θ m+M A body of mass M and radius R is rolling horizontally without slipping with speed v. It then rolls up a hill to a maximum height h. If h = 5v2/6g, what is the M.I. of the body ? 2 1 (b) MR2 a) MR2 2 3 3 2 2 (d) MR2 (c) MR 4 5 A boy and a man carry a uniform rod of length l horizontally 1 in such a way that the boy gets th of the load. If the boy is 4 at one end of the rod, then the distance of the man from the other end is 3l l 2l l (b) (c) (d) (a) 3 3 4 4 A carpenter has constructed a toy as shown in the adjoining fig.

9.

If the density of the material of the sphere is 12 times that of cone, the position of the centre of mass of the toy is given by : (a) at a distance of 2R from O (b) at a distance of 3R from O (c) at a distance of 4R from O (d) at a distance of 5R from O A certain bicycle can go up a gentle incline with constant speed when the frictional force of ground pushing the rear wheel if F2 = 4 N. With what force F1 must the chain pull on the sprocket wheel if R1 = 5 cm and R2 = 30 cm ? (a) 4 N (b) 24 N 35 N c) 140 N d) 4 A heavy disc is thrown on a horizontal surface in such a way that it slides with a speed V0 initially without rolling. It will start rolling without slipping when its speed reduces to : V 2V0 (b) (a) 0 2 3 5V0 3V0 (c) d) 5 7 A heavy particle hanging from a string of length l is projected horizontally with speed

gl . The speed of the particle at the

point where the tension in the string equals the weight of the particle is (a)

(2gl)

(b)

(3gl)

(c)

(gl / 2)

(d)

(gl / 3)

10. A mass m is supported by a massless string wound around a uniform cylinder of mass m and radius R, Fig. With what acc. will the mass fall on release ? (a) 2g/3 b) g/2 c) g (d) 4g/3 11. A particle of mass m = 5 is moving with a uniform speed v = 3√2 in the XOY plane along the line y = x + 4. The magnitude of the angular momentum about origin is (a) zero b) 60 unit (c) 7.5 unit (d) 40 √2 12. A particle of mass m is rotating in a plane in circular path of radius r. Its angular momentum is L. The centripetal force acting on the particle is

(a)

L2 mr

b)

the normal of inertia of the same cylinder about an axis passing through its centre and normal to its length l then :

L2 m r2

(a) L = R (b) L = 3 R R (c) L = (d) L = 0 3 13. A particle originally at rest at the highest point of a smooth 19. A uniform solid right circular cone of base radius r is joined to a uniform solid hemisphere of radius r and of the same vertical circle is slightly displaced. It will leave the circle at a density, so as to have a common face. The centre of gravity vertical distance h below the highest point, such that of the composite solid lies on the common face. The height of (a) h = R b) h = 2R the cone is (c) h = R/2 (d) h = R/3 14. A rigid body rotates about a fixed axis with variable angular (b) r/ 6 a) 3r/ 2 velocity equal to α - βt at time t where α and β are constants. c) 4r (d) r 3 The angle through which it rotates before it comes to rest is : 20. Angular momentum and areal velocity of a body of mass m α 2 − β2 α2 (b) (a) are related as 2α 2β a) L = 2m × areal velocity 2 2 b) L = 2π × areal velocity α(α − β) α −β (c) (d) c) Areal velocity = 2mL 2 2β (d) 2m = 2 × areal velocity 15. A ring rolls down, starting from rest, down an inclined plane 21. Consider the rotation of a rod of mass m and length l from of length l and inclination θ. The velocity of the centre of position AB to AB/. mass of the ring at the mid-point of the inclined plane is 2

L d) . mr

L2 c) mr 3

gl sin θ

(a)

(b)

4 gl sin θ 7 16. A thick walled hollow sphere has outer radius R. It rolls down an inclined plane without slipping and its speed at the bottom is v. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom will be 5v/4. What is the radius of gyration of the sphere? R R (b) (a) 2 2 2gl sin θ

(c)

A

gl sin θ 2

(d)

B

B’

Which of the following is correct ? a) Weight of the rod is lowered by

l . 2

b) Loss of gravitational potential energy is (c) Angular velocity is

1 mgl. 2

3g . l

3R 3R (d) 4 4 ml2ω2 (d) Rotational kinetic energy is 17. A uniform circular disc of radius a is taken. A circular portion 3 of radius b has been removed from it as shown in the fig. 22. Fig. shows a thin metallic triangular sheet ABC. (c)

A a

l

B

b l

O

c

f the centre of hole is at a distance c from the centre of the disc, the distance x2 of the centre of the mass the remaining part from the initial centre of mass O is given by :

πb 2

C

The mass of the sheet is M. The moment of inertia of the sheet about side AC is :

− cb 2

Ml 2 Ml 2 Ml 2 Ml 2 (b) (c) (d) (a − b ) (a − b ) 18 12 6 4 23. From a given sample of uniform wire, two circular loops P 2 2 πc πa (c) 2 (d) and Q are made, P of radius r and Q of radius nr. If the M.I. (a − b 2 ) (c 2 − b 2 ) of Q about its axis is 4 times that of P about its axis 18. A uniform cylinder has a radius R and length L. If the (assuming wire diameter much smaller, than either radius), moment of inertia of this cylinder about an axis passing the value of n is : through its centre and normal to its circular face is equal to a)

2

2

(b)

2

2

(a)

(a) (4)2/3 b) (4)1/3 1/2 d) (4)1/4 c) (4) 24. If a disc slides from top to bottom of an inclined plane, it

Z

t2 takes time t1. If it rolls, it takes time t2. Now, 22 is t1 Y 2 1 (a) (b) x 2 3 2 3 The moment of inertia of this system about Z-axis is : (d) (c) 5 2 ML2 2ML2 4ML2 5ML2 (b) (c) (d) (a) 25. If a sphere is rolling, the ratio of the translational energy to 3 3 3 3 total kinetic energy is given by 32. Two discs, each of mass 1 kg and radius metre, are 2 a) 7:10 b) 2:5 placed parallel to each other and separated by a distance of (c) 10:7 d) 5:7 26. If the earth is treated as a sphere of radius R and mass M, its 2 metre. angular momentum about the axis of its diurnal rotation with Y period T is

2πMR 2 4πMR 2 MR 2T πMR 3 (b) (c) (d) 5T T 2π T 27. In fig. (a), a metre stick, half of which is wood and the other half steel is pivoted at the wooden end at O and a force F is applied to the steel end A. (a)

O Wood

Steel

A

F

Wood A/

F

(a ) O/ Steel

Y'

YY’ is an axis which is perpendicular to the line joining the centers of the two discs and is mid-way between the two discs. The moment of inertia of the system about YY’ is (b) 2 kg m2 (a) 1 kg m2 2 (d) 8 mg r12 (c) 4 kg m 33. Two thin discs each of mass M and radius r metre are attached as shown in fig., to form a rigid body.

b A In fig. (b) the stick is pivoted at the steel end at O’ and B the same force F is applied at the wooden end at A’. The angular acceleration: (a) in (a) is greater than in (b) The rotational inertia of this body about an axis (b) in (b) is greater than in (a) perpendicular to the plane of disc B and passing through its (c) is equal both in (a) and (b) centre is : (d) none of the above b) 3Mr2 a) 2Mr2 2 28. Moment of inertia of a uniform circular disc about a diameter d) 5Mr2 c) 4Mr is I. Its moment of inertia about an axis ⊥ to its plane and 34. Two uniform thin identical rods AB and CD each of mass M passing through a point on its rim will be and length L are joined so as to form a cross as shown. (a) 5I (b) 3I (c) 6I (d) 4I C F 29. The mean kinetic energy of a particle of mass m moving with constant force in any interval of time will be …. where ω1 A B and ω2 are the initial and final velocities. m m 2 (ω1 + ω1ω 2 + ω 22 ) (b) (ω1 + ω 2 ) 2 (a) E 6 2 D m m 2 2 The moment of inertia of the cross about a bisector line (ω1 + ω 2 ) (ω1 − ω 2 ) (d) (c) 3 6 EF is : 30. The rotational kinetic energy of a body is E and its moment ML2 ML2 ML2 ML2 of inertia is 1. The angular momentum is (b) (c) (d) (a) 6 4 12 3 (a) EI (b) 2 IE 35. When a ceiling fan is switched off, its angular velocity falls to half while it makes 36 rotations. How many more rotations (d) E/I. (c) 2 EI will it make before coming to rest ? (Assume uniform angular 31. Three thin rods each of length L and mass M are placed along retardation) X, Y and Z-axes in such a way that one end of each of the (a) 36 (b) 24 rods is at the origin. (c) 18 (d) 12

36. The moment of inertia of a uniform thin rod of length L and mass M about an axis passing through a point at a distance of L/3 from one of its ends and perpendicular to the rod is :

38.

39.

40.

41.

(a)

g L

(b)

1 2

g L

3 g g 7 ML2 ML2 (c) (d) 2 (b) 2 L L 48 1 42. A uniform rod of mass M and length L is pivoted at 2 ML ML2 one end such that it can rotate in a vertical plane. (d) (c) 9 3 There is negligible friction at the pivot. The free end Two thin discs each of mass M and radius r metre are of the rod is held vertically above the pivot and then attached as shown in fig., to form a rigid body. The released. The angular acceleration of the rod when it rotational inertia of this body about an axis makes an angle θ with the vertical is : perpendicular to the plane of disc B and passing g (a) g sin θ (b) sin θ through its centre is : l (b) 3Mr 2 (a) 2Mr 2 3 g (d) 5Mr 2 (c) 4Mr 2 sin θ (d) 6 gl sin θ (c) 2l A uniform disc of mass M and radius R is mounted on an axle supported in frictionless bearings. A light cord 43. A uniform cube of side a and mass m rests on a rough horizontal table. A horizontal force F is applied is wrapped around the rim of the disc and a steady normal to one of the faces at a point that is directly downward pull T is exerted on the cord. The angular 3a acceleration of the disc is : above the base. above the centre of face, at a height 4 MR T (b) (a) The minimum value of F for which the cube begins to T MR tilt about the edge is (assume that the cube does not MR 2T (d) (c) slide) : 2T MR 2mg mg A body is rolling without slipping on a horizontal (b) (a) 4 3 surface and its rotational kinetic energy is equal to the 3 mg translational kinetic energy. The body is : (d) mg (c) (a) disc (b) sphere 4 (c) cylinder (d) ring 44. A cubical block of mass M and edge a slides down a A particle is mass m is projected with a velocity v rough inclined plane of inclination θ with a uniform making an angle of 45° with the horizontal. The velocity. The torque of the normal force on the block magnitude of angular momentum of the projectile about its centre has a magnitude : about an axis of projection when the particle is at (a) zero (b) Mga maximum height h is : Mga sin θ (c) Mga sin θ (d) mv3 2 (a) zero (b) 45. A rigid spherical body is spinning around an axis 4 2g without any external torque. Due to change in mv 2 temperature, the volume increases by 1%. Its angular (c) (d) m 2gh 3 speed : 2g (a) will increase approximately by 1% A rigid body is made of three identical thin rods, each (b) will decrease approximately by 1% of length L fastened together in the form of letter H. (c) will decrease approximately by 0.67% The body is free to rotate about a horizontal axis that (d) will decrease approximately by 0.33% runs along the length of one of the legs of the H. The body is allowed to fall from rest from a position in which the plane of H is horizontal. What is the angular (a)

37.

speed of the body when the plane of H is vertical ?