Rotational Motion

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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. Answer: 4. 108 days Key idea: no external torque, so angular momentum L=Iω is conserved. For a uniform sphere, I=2/5MR² . Mass stays the sam... Angular momentum and its conservation 1 answer
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 m/s^2) Answer: 2. 100√(3) N Key idea: the wall is smooth, so the wall force N is horizontal. For equilibrium, floor friction f equals N , and torqu... Equilibrium of rigid bodies, rigid body rotation, equations of rotational motion 1 answer
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 : Answer: 3. 7/57 Key idea: moment of inertia of the remaining part = full sphere - removed sphere, using the parallel-axis theorem for the cu... Moment of inertia, radius of gyration, MI of simple objects 1 answer
A wheel of a bullock cart is rolling on a level road as shown in the figure below. If its linear speed is v 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)? Answer: 3. Point P moves faster than point Q . Key idea: In pure rolling, each point has speed = (speed of centre) + (rotational speed about... Equilibrium of rigid bodies, rigid body rotation, equations of rotational motion 1 answer
The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is 2400 g cm^2. The length of the 400 g rod is nearly : Answer: 2. 8.5 cm Key idea: For a thin rod about its midpoint and perpendicular to its length, I=ML²/12 . L²=12I/M=12×2400/400=72 , so L=√(7... Moment of inertia, radius of gyration, MI of simple objects 1 answer
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 : Answer: Dropped in the official answer key: full marks to every candidate Radius of gyration is k=√(I/M) . Solid sphere about its axis: I=2/... Moment of inertia, radius of gyration, MI of simple objects 1 answer
Two objects of mass 10 kg and 20 kg respectively are connected to the two ends of a rigid rod of length 10 m with negligible mass. The distance of the center of mass of the system from the 10 kg mass is : Answer: 3. 20/3 m Place the 10 kg mass at the origin and the 20 kg mass at 10 m. x_cm=10(0)+20(10)/10+20=200/30=20/3\ m The centre of mass l... Centre of mass of two-particle system and rigid body 1 answer
The angular speed of a fly wheel moving with uniform angular acceleration changes from 1200 rpm to 3120 rpm in 16 seconds. The angular acceleration in rad/s^2 is : Answer: 3. 4π Convert rpm to rad/s using ω=2π N/60 . Δ N=3120-1200=1920 rpm, so Δω=2π×1920/60=64π\ rad/s . α=Δω/t=64π/16=4π\ rad/s² . Option... Comparison of linear and rotational motion 1 answer
The ratio of the radius of gyration of a thin uniform disc about an axis passing through its centre and normal to its plane to the radius of gyration of the disc about its diameter is : Answer: 3. √(2) : 1 Radius of gyration is k=√(I/M) . For a disc, about the normal axis through the centre I=1/2MR² ; about a diameter I=1/4M... Moment of inertia, radius of gyration, MI of simple objects 1 answer
From a circular ring of mass 'M' and radius 'R' an arc corresponding to a 90° sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring i… Answer: 1. 3/4 Key idea: moment of inertia of a ring about its axis depends only on mass and radius, since every part is at distance R. Remo... Moment of inertia, radius of gyration, MI of simple objects 1 answer
A uniform rod of length 200 cm and mass 500 g is balanced on a wedge placed at 40 cm mark. A mass of 2 kg is suspended from the rod at 20 cm and another unknown mass 'm' is suspended from the rod at 160 cm mark as shown in the figure. Find the value… Answer: 4. 1/12 kg Key idea: take torques about the wedge (40 cm mark). The rod's weight acts at its centre (100 cm). Anticlockwise (left):... Equilibrium of rigid bodies, rigid body rotation, equations of rotational motion 1 answer
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 : Answer: 3. 67 cm Take the 5 kg mass as the origin. Centre of mass: x_cm=5(0)+10(100)/5+10=1000/15≈67\ cm It lies closer to the heavier mass,... Centre of mass of two-particle system and rigid body 1 answer
Find the torque about the origin when a force of 3j N acts on a particle whose position vector is 2k m. Answer: 3. -6i N m Torque vecτ=vec r×vec F . vecτ=2hat k×3hat j=6(hat k×hat j)=6(-hat i)=-6hat i\ N m Since hat j×hat k=hat i , reversing th... Moment of force, torque 1 answer
A solid cylinder of mass 2 kg and radius 4 cm is rotating about its axis at the rate of 3 rpm. The torque required to stop after 2π revolutions is : Answer: 1. 2×10⁻⁶ N m Find the angular deceleration from the rotation kinematics, then use τ=Iα . ω₀=3 rpm =3×2π/60=0.1π rad/s. Angle turned... Moment of force, torque 1 answer
A disc of radius 2 m and mass 100 kg rolls on a horizontal floor. Its centre of mass has speed of 20 cm/s. How much work is needed to stop it ? Answer: 1. 3 J To stop the disc we must remove its total kinetic energy, which is translational plus rotational. K=1/2mv²+1/2Iω² with I=1/2m... Comparison of linear and rotational motion 1 answer
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 of the following physical quantities would remain constant for the sphere ? Answer: 4. Angular momentum The sphere is in free space, so there is no external torque and angular momentum L=Iω is conserved. When the rad... Angular momentum and its conservation 1 answer
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 Answer: 3. 5 : 7 For a solid sphere rolling without slipping, K_t=1/2mv² and K_r=1/2Iω²=1/2·2/5mR²·v²/R²=1/5mv² . K_t/K_t+K_r=1/2/1/2+1/5=5/... Comparison of linear and rotational motion 1 answer
Three objects, A : (a solid sphere), B : (a thin circular disk) and C : (a circular 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 th… Answer: 1. W_C>W_B>W_A Work needed to stop each body equals its rotational kinetic energy, W=1/2Iω² . Same M , R , ω , so W∝ I . Solid spher... Moment of inertia, radius of gyration, MI of simple objects 1 answer
The moment of the force, F=4 i+5 j-6 k at (2, 0, -3), about the point (2, -2, -2), is given by Answer: 4. -7hat i-4hat j-8hat k Moment (torque) about a point: vecτ=vec r×vec F , where vec r goes from the point to the point of applicati... Moment of force, torque 1 answer
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 ω_1 and ω_2. They are brought into contact face to face coinciding the axis of rotation. The e… Answer: 1. 1/4 I(ω₁-ω₂)² Key idea: angular momentum is conserved when the discs join. Iω₁+Iω₂=2Iω⇒ω=ω₁+ω₂/2 . Loss =1/2Iω₁²+1/2Iω₂²-1/2(2I)ω... Angular momentum and its conservation 1 answer
Which of the following statements are correct ?(a) Centre of mass of a body always coincides with the centre of gravity of the body.(b) Centre of mass of a body is the point at which the total gravitational torque on the body is zero.(c) A couple on… Answer: 4. (b) and (d) Key idea: check each statement. (a) False: centre of mass and centre of gravity coincide only in a uniform gravitatio... Centre of mass of two-particle system and rigid body 1 answer
A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N ? Answer: 2. 25 rad/s ² Key idea: τ=Iα , with τ=FR and I=MR² for a hollow cylinder. τ=30×0.4=12 N m, I=3×0.4²=0.48 kg m ² . α=12/0.48=25 rad/s... Equilibrium of rigid bodies, rigid body rotation, equations of rotational motion 1 answer
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 equal kinetic energy of rotation. If L_A and L_B be their angular momenta respectively, then Answer: 2. L_B>L_A Express angular momentum through kinetic energy: K=L²/2I , so L=√(2IK) . The kinetic energies are equal, so L∝sqrt I . Si... Comparison of linear and rotational motion 1 answer
A light rod of length l has two masses m_1 and m_2 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 Answer: 4. m₁m₂/m₁+m₂l² Take the two masses at distances x₁ and x₂ from the centre of mass, with x₁+x₂=l and m₁x₁=m₂x₂ . This gives x₁=m₂ l/... Moment of inertia, radius of gyration, MI of simple objects 1 answer

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