193
Rotational Mechanics
OBJECTIVE I
mass m and length l is
1. Let
A be a unit vector along the axis of rotation of a 9. One end of a uniform rod of horizontal surface
clamped. The rod lies on a smooth
svely rotating body and B be a unit vector along the about the clamped end at a uniform
velocity of a particle P of the body away from the axis. and rotates on it
clamp on
is angular velocity ). The force exerted by the
The value of A.B the rod has a horizontal component
(a) 1 (b) -1 (c) 0 (d) None of these. 1
Abody is uniformly rotating about an axis fixed in an (a) mo (b) zero (c) mg (d) 2
mo1.
inertial frame of reference. Let A be a unit vector along smooth
10. A uniform rod is kept vertically on a horizontal
the axis of rotation and B be the unit vector along the surface at a point 0. If it is rotated slightly and released,
resultant force on a particle P of the body away from it falls down on the horizontal surface. The lower end
the axis. The value of A.Bis will remain
(a)1 (b) -1 (c) 0 (d) none of these. (a) at 0 (b) at a distance less than /2 from O
Aparticle moves with a constant velocity parallel to the (c) at a distance l/2 from O
K-axis. Its angular momentum with respect to the origin (d) at a distance larger than l/2 from 0.
(a) is zero (b) remains constant 11, A circular disc A of radius r is made from an iron plate
(c) goes on increasing (d) goes on decreasing. of thickness t and another circular disc B of radius 4r
4. Abody is in pure rotation. The linear speed of is made from an iron plate of thickness t/4. The relation
particle, the distance r of the particle from the axis and between the moments of inertia I, and I, is
the angular velocity o of the body are related as (a) I, >I (b)I, =I (c) I, <I,
)= Thus (d) depends on the actual values of t and r.
(a)o I
12. Equal torques act on the discs A and B of the previous
problem, initially both being at rest. At a later instant,
(c) o = 0 (d) o is independent of r. the linear speeds of a point on the rim of A and another
point on the rim of B are V, and vg respectively. We
5. Figure (10-Q3) shows a small wheel fixed coaxially on a have
bigger one of double the radius. The system rotates (a) va> VB (b) V = Ug (c) v,< Up
about the common axis. The strings supporting A and B (a) the relation depends on the actual magnitude of the
do not slip on the wheels. If x and y be the distances
travelled by A and B in the same time interval, then torques.
(a) x=2y (b) x=y (c) y =2 x (d) none of these. 13. A closed cylindrical tube containing some water (not
flling the entire tube) lies in a horizontal plane. If the
tube is rotated about a perpendicular bisector, the
moment of inertia of water about the axis
(a) increases (b) decreases (c) remains constant
(d) increases if the rotation is clockwise and decreases
if it is anticlockwise.
14. The moment of inertia of a uniform semicircular wire of
mass M and radius r about a line perpendicular to the
Figure 10-Q3 plane of the wire through the centre is
1
(a) M 2 (b) M2 (c) 1 (d) 2
6. A body is rotating uniformly about a vertical axis fixed
in an inertial frame. The resultant force on a particle of 15. Let I, and I, be the moments of inertia of two bodies of
the body not on the axis is identical geometrical shape, the first made of aluminium
(a) vertical (b) horizontal and skew with the axis and the second of iron.
(c) horizontal and intersecting the axis (a) I, <I, (b) I, = I, (c) I, > I,
(d) none of these. (d) relation between I, and I, depends on the actual
7. A body is rotating nonuniformly about a vertical axis shapes of the bodies.
fixed in an inertial frame. The resultant force on a 16. A body having its centre of mass at the origin has three
particle of the body not on the axis is of its particles at (a,0,0), (0,a,0), (0,0,a). The moments
(a) vertical (b) horizontal and skew with the axis of inertia of the body about the X and Y axes are
(c) horizontal and intersecting the axis 0-20 kg-m 2 each. The moment of inertia about the
(d) none of these. Z-axis
2
D Let F be a force acting on a particle having position (a) is 0-20 kg-m (b) is 040 kg-m
vector r. Let T be the torque of this force about the (c) is 0:20V2 kg-m
origin, then (d) cannot be deduced with this information.
(a) r,r=0 and F.T=0 (b) r.r=0 but F.r+0 17. A cubical block of mass M and edge a slides down a
(c) r.r0 but F.T=0 (d) r.r0 and F,r0. rough inclined plane of inclination with a uniform
Rotational Mechanics
OBJECTIVE I
mass m and length l is
1. Let
A be a unit vector along the axis of rotation of a 9. One end of a uniform rod of horizontal surface
clamped. The rod lies on a smooth
svely rotating body and B be a unit vector along the about the clamped end at a uniform
velocity of a particle P of the body away from the axis. and rotates on it
clamp on
is angular velocity ). The force exerted by the
The value of A.B the rod has a horizontal component
(a) 1 (b) -1 (c) 0 (d) None of these. 1
Abody is uniformly rotating about an axis fixed in an (a) mo (b) zero (c) mg (d) 2
mo1.
inertial frame of reference. Let A be a unit vector along smooth
10. A uniform rod is kept vertically on a horizontal
the axis of rotation and B be the unit vector along the surface at a point 0. If it is rotated slightly and released,
resultant force on a particle P of the body away from it falls down on the horizontal surface. The lower end
the axis. The value of A.Bis will remain
(a)1 (b) -1 (c) 0 (d) none of these. (a) at 0 (b) at a distance less than /2 from O
Aparticle moves with a constant velocity parallel to the (c) at a distance l/2 from O
K-axis. Its angular momentum with respect to the origin (d) at a distance larger than l/2 from 0.
(a) is zero (b) remains constant 11, A circular disc A of radius r is made from an iron plate
(c) goes on increasing (d) goes on decreasing. of thickness t and another circular disc B of radius 4r
4. Abody is in pure rotation. The linear speed of is made from an iron plate of thickness t/4. The relation
particle, the distance r of the particle from the axis and between the moments of inertia I, and I, is
the angular velocity o of the body are related as (a) I, >I (b)I, =I (c) I, <I,
)= Thus (d) depends on the actual values of t and r.
(a)o I
12. Equal torques act on the discs A and B of the previous
problem, initially both being at rest. At a later instant,
(c) o = 0 (d) o is independent of r. the linear speeds of a point on the rim of A and another
point on the rim of B are V, and vg respectively. We
5. Figure (10-Q3) shows a small wheel fixed coaxially on a have
bigger one of double the radius. The system rotates (a) va> VB (b) V = Ug (c) v,< Up
about the common axis. The strings supporting A and B (a) the relation depends on the actual magnitude of the
do not slip on the wheels. If x and y be the distances
travelled by A and B in the same time interval, then torques.
(a) x=2y (b) x=y (c) y =2 x (d) none of these. 13. A closed cylindrical tube containing some water (not
flling the entire tube) lies in a horizontal plane. If the
tube is rotated about a perpendicular bisector, the
moment of inertia of water about the axis
(a) increases (b) decreases (c) remains constant
(d) increases if the rotation is clockwise and decreases
if it is anticlockwise.
14. The moment of inertia of a uniform semicircular wire of
mass M and radius r about a line perpendicular to the
Figure 10-Q3 plane of the wire through the centre is
1
(a) M 2 (b) M2 (c) 1 (d) 2
6. A body is rotating uniformly about a vertical axis fixed
in an inertial frame. The resultant force on a particle of 15. Let I, and I, be the moments of inertia of two bodies of
the body not on the axis is identical geometrical shape, the first made of aluminium
(a) vertical (b) horizontal and skew with the axis and the second of iron.
(c) horizontal and intersecting the axis (a) I, <I, (b) I, = I, (c) I, > I,
(d) none of these. (d) relation between I, and I, depends on the actual
7. A body is rotating nonuniformly about a vertical axis shapes of the bodies.
fixed in an inertial frame. The resultant force on a 16. A body having its centre of mass at the origin has three
particle of the body not on the axis is of its particles at (a,0,0), (0,a,0), (0,0,a). The moments
(a) vertical (b) horizontal and skew with the axis of inertia of the body about the X and Y axes are
(c) horizontal and intersecting the axis 0-20 kg-m 2 each. The moment of inertia about the
(d) none of these. Z-axis
2
D Let F be a force acting on a particle having position (a) is 0-20 kg-m (b) is 040 kg-m
vector r. Let T be the torque of this force about the (c) is 0:20V2 kg-m
origin, then (d) cannot be deduced with this information.
(a) r,r=0 and F.T=0 (b) r.r=0 but F.r+0 17. A cubical block of mass M and edge a slides down a
(c) r.r0 but F.T=0 (d) r.r0 and F,r0. rough inclined plane of inclination with a uniform
