1. A uniform rod of length 200 cm and mass 500 g is balanced on a wedgeplaced at 40 cm mark. A mass of 2 kg is suspended from the rod at 20cm and another unknown mass 'm' is suspended from the rod at 160 cmmark as shown in the figure. Find the value of 'm' such that the rod is inequilibrium. (g = 10 m/s2)

(1).1 / 2 kg

(2). 1 / 3 kg

(3). 1 / 6 kg

(4). 1 / 12 kg

2. The radius of gyration of a solid sphere of mass 5 kg about X Y is 5 m as shown in figure. The radius of the sphere is \(\frac{5x}{\sqrt{7}}\) m then the value of x is

(1).5

(2). \(\sqrt{2}\)

(3). \(\sqrt{3}\)

(4). \(\sqrt{5}\)

3. The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is 2400 gcm². The length of the 400g rod is nearly:

(1).8.5

(2). 17.5

(3). 20.7

(4). 72.0

4. A wheel of a bullock cart is rolling on a level road as shown in the figure below. If its linear speed is ν in the direction shown, which one of the following options is correct (P and Q are any highest and lowest points on the wheel, respectively)?

(1).Point P moves slower than point Q

(2). Point P moves faster than point Q

(3). Both the points P and Q move with equal speed

(4). Point P has zero speed

5. Two particles A and B initially at rest, move towards each other under mutual force of attraction. At an instance when the speed of A is v and speed of B is 3v, the speed of centre of mass is:

(1).2v

(2). zero

(3). v

(4). 4v

6. A constant torque of 100Nm turns a wheel of moment of inertia 300kgm about an axis passing through its centre. Starting from rest, its angular velocity after 3 s is :-

(1).1rad∕ s

(2). 5rad∕ s

(3). 10rad∕ s

(4). 15rad∕ s

7. The ratio of radius of gyration of a solid sphere of mass M and radius R about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is

(1).5:3

(2). 2:5

(3). 5:2

(4). None of the above

8. The angular acceleration of a body, moving along the circumference of a circle, is

(1).Along the radius towards the centre

(2). Along the tangent to its position

(3). Along the axis of rotation

(4). Along the radius, away from centre

9. A shell of mass m is at rest initially. It explodes into three fragments having mass in the ratio 2 : 2 : 1.If the fragments having equal mass flyoff along mutually perpendicular directions with speed v, the speed ofthe third (lighter) fragment s

(1).v

(2). \(\sqrt{2}\)v

(3). 2\(\sqrt{2}\)v

(4). 3\(\sqrt{2v}\)

10. The angular speed of a fly wheel moving with uniform angular acceleration changes from 1200rpm to 3120 rpm in 16 seconds. The angular acceleration in rad ∕ s² is

(1).2π

(2). 4π

(3). 12π

(4). 104π

11. Two objects of mass 10kg and 20kg respectively are connected to the two ends of a rigid rod of length 10m with negligible mass. The distance of the center of mass of the system from the 10kg mass is

(1).\(\frac{10}{3}\)m

(2). \(\frac{20}{3}\)m

(3). 10m

(4). 5m

12. The ratio of the radius of gyration of a thin uniform disc about an axispassing through its centre and normal to its plane to the radius ofgyration of the disc about its diameter is

(1).2 : 1

(2). \(\sqrt{2}\) : 1

(3). 4 : 1

(4). 1 : \(\sqrt{2}\)

13. An energy of 484J is spent in increasing the speed of a flywheel from60rpm to 360rpm. The moment of inertia of the flywheel is

(1).0.07 kg − m2

(2). 0.7 kg − m 2

(3). 3.22 kg − m 2

(4). 30.8 kg − m 2

14. Given below are two statements: one is labelled as Assertion (A) and theother is labelled as Reason (R).Assertion (A) :When a fire cracker (rocket) explodes in mid air, its fragments fly insuch a way that they continue moving in the same path, which the firecracker would have followed, had it not exploded.Reason (R):Explosion of cracker (rocket) occurs due to internal forces only and noexternal force acts for this explosion.In the light of the above statements, choose the most appropriateanswer from the options given below

(1). (A) is not correct but (R) is correct

(2). Both (A) and (R) are correct and (R) is the correct explanation of (A)

(3). Both (A) and (R) are correct but (R) is not the correct explanation of (A)

(4). (A) is correct but (R) is not correct

15. From a circular ring of mass ' M and radius ' R ' an arc correspondingto a 90° sector is removed. The moment of inertia of the remaining partof the ring about an axis passing through the centre of the ring andperpendicular to the plane of the ring is ' K times ′M R2'. Then thevalue of ' K is

(1).\(\frac{3}{4}\)

(2). \(\frac{7}{8}\)

(3). \(\frac{1}{4}\)

(4). \(\frac{1}{8}\)

16. A solid cylinder of mass 2kg and radius 4cm is rotating about its axis atthe rate of 3 rpm. The torque required to stop it after 2π revolutions is

(1).2 × 10 6N m

(2). 2 × 10 −6N m

(3). 2 × 10 −3N m

(4). 12 × 10 −4N

17. A disc of radius 2m and mass 100kg rolls on a horizontal floor. Itscentre of mass has speed of 20cm ∕ s. How much work is needed to stopit?

(1).1 J

(2). 3 J

(3). 30 kJ

(4). 2J

18. A solid cylinder of mass 2kg and radius 50cm rolls up an inclined planeof angle inclination \(30^\circ\). The centre of mass of cylinder has speed of 4m ∕ s. The distance travelled by the cylinder on the incline surface will be (Take g = 10m ∕ s2)

(1).2.2m

(2). 1.6m

(3). 1.2m

(4). 2.4m

19. A solid sphere is in rolling motion. In rolling motion a body possesses translational kinetic energy (K_t) as well as rotational kinetic energy(K_r) simultaneously. The ratio K_t: (K_t + K _r) for the sphere is

(1).7:10

(2). 5:7

(3). 10:7

(4). 2:5

20. A solid sphere is rotating freely about its symmetry axis in free space.The radius of the sphere is increased keeping its mass same. Which ofthe following physical quantities would remain constant for the sphere?

(1).Angular velocity

(2). moment of inertia

(3). Rotational kinetic energy

(4). Angular momentum

21. Three objects, A (a solid sphere), B (a thin circular disk) and C (acircular ring), each have the same mass M and radius R. They all spin with the same angular speed ω about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation

(1).W C > W B > W A

(2). W A > W B > W C

(3). W B > W A > W C

(4). W A > W C > W B

22. The moment of the force, \(\vec{F} = 4\hat{i} + 5\hat{j} − 6\hat{k}\) at (2, 0, −3), about the point (2, −2, −2), is given by

(1).\(- 8\hat{i} − 4\hat{j} − 7\hat{k}\)

(2). \(−4\hat{i} −\hat{j} − 8\hat{k}\)

(3). \(−7\hat{i} − 8\hat{j} − 4\hat{k}\)

(4). \(−7\hat{i} − 4\hat{j} − 8\hat{k}\)

23. A rope is wound around a hollow cylinder of mass 3 kg and radius 40cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N?

(1). 0.25 rad s⁻²

(2). 25 rad s⁻²

(3). 5 ms⁻²

(4). 25 ms⁻²

24. Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ω₁ and ω₂.They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is

(1).\(\frac{1}{4} I \left(ω_1 − ω_2\right)^2\)

(2). \(I \left(ω_1 − ω_2\right)^2\)

(3). \(\frac{1}{8} I \left(ω_1 − ω_2\right)^2\)

(4). \(\frac{1}{2} I \left(ω_1 + ω_2\right)^2\)

25. Which of the following statements are correct?
(1) Centre of mass of a body always coincides with the centre of gravity of the body.
(2) Centre of mass of a body is the point at which the total gravitational torque on the body is zero.
(3) A couple on a body produces both transnational and rotational motion in a body.
(4) Mechanical advantage greater than one means that small effort can be used to lift a large load.

(1).(1) and (2)

(2). (2) and (3)

(3). (3) and (4)

(4). none of the above

26. Two rotating bodies A and B of masses m and 2m with moments of inertia I_A and I_B (I_B > I _A) have a equal kinetic energy of rotation. If L_A and L_B be their angular momentum respectively, then

(1).\(L_A = \frac{L_B}{2}\)

(2). \(L_A = 2L_B\)

(3). \(L_B > L_A\)

(4). \(L_A > L_B\)

27. A solid sphere of mass m and radius R is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation \(\frac{E_{sphere}}{E_{cylinder}}\) will be

(1).2 : 3

(2). 1 : 5

(3). 1 : 4

(4). 3 : 1

28. A light rod of length l has two masses m₁ and m₂ attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is

(1).\(\frac{m_1m_2}{m_1 + m_2}I^2\)

(2). \(\frac{m1 + m2}{m1m2}I^2\)

(3). \((m_ 1+ m_2)I^2\)

(4). \(\sqrt{m_1m_2I^2}\)

29. A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

(1).Both reach at the same time

(2). Depends on their masses

(3). Disk

(4). Sphere

30. From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?

(1).\({11 MR^2}/{32}\)

(2). \({9 MR^2}/{32}\)

(3). \({15 MR^2}/{32}\)

(4). \({13 MR^2}/{32}\)

31. A uniform circular disc of radius 50 cm at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of 2.0 rad s⁻². Its net acceleration in s⁻² at the end of 2.0 S is approximately

(1).6.0

(2). 3.0

(3). 8.0

(4). 7.0

32. Point masses m₁ and m₂ are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ω₀ is minimum, is given by

(1).x = \(\frac{m_2}{m_1}\)L

(2). x = \(\frac{m_2L}{m_1 + m_2}\)

(3). x = \(\frac{m_1L}{m_1 + m_2}\)

(4). x = \(\frac{m_1}{m_2}\)L

33. An automobile moves on a road with a speed of 54 kmh⁻¹.The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kgm². If the vehicle is brought to rest in 15 S, the magnitude of average torque transmitted by its brakes to the wheel is

(1).10.86 kgm²s⁻²

(2). 2.86 kgm²s⁻²

(3). 6.66 kgm²s−2

(4). 8.58 kgm²s⁻²

34. A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A.The normal reaction on A is

(1).\(\frac{W (d − x)}{x}\)

(2). \(\frac{W (d − x)}{d}\)

(3). \(\frac{Wx}{d}\)

(4). \(\frac{Wd}{x}\)

35. Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis X X ' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about X X ' axis is

(1).\(\frac{16}{5}\)mr2

(2). 4mr2

(3). \(\frac{11}{5}\)mr2

(4). 3mr2

36. A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate aboutthe horizontal axis. A mass less string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s⁻² is

(1).25 N

(2). 50 N

(3). 78.5 N

(4). 157 N

37. The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is

(1).5 : 7

(2). 2 : 3

(3). 2 : 5

(4). 7 : 5

38. Two particles of mass 5 kg and 10 kg respectively are attached to the two ends of a rigid rod of length 1 m with negligible mass.The centre of mass of the system from the 5 kg particle is nearly at a distance of

(1).50 cm

(2). 67 cm

(3). 80 cm

(4). 33 cm

39. Find the torque about the origin when a force of \(3\, \hat{j}\, \text{N}\) acts on a particle whose position vector is \(2\, \hat{k}\, m\).

(1).\(6\, \hat{j}\, \text{Nm}\)

(2). \(-6\, \hat{i}\, \text{Nm}\)

(3). \(6\, \hat{k}\, \text{Nm}\)

(4). \(6\, \hat{i}\, \text{Nm}\)

40. The sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.

(1).108 days

(2). 100 days

(3). 105 days

(4). 115 days

41. A uniform rod of mass 20 kg and length 5 m leans against a smooth vertical wall making an angle of 60° with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (take\( g\, =\, 10 \text{m/s}^2\))

(1).\(200\, \sqrt{3}\, \text{N}\)

(2). \(100\, \sqrt{N}\)

(3). \(100 \, \sqrt{3}\, \text{N}\)

(4). \(200\, \text{N}\)

42. A sphere of radius R is cut from a larger solid sphere of radius 2R as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is :

(1).\(\displaystyle \frac{7}{64}\)

(2). \(\displaystyle \frac{7}{8}\)

(3). \(\displaystyle \frac{7}{40}\)

(4). \(\displaystyle \frac{7}{57}\)