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The correct choice is (C) 12.5Nm

For explanation: w = w0 + a*dt OR 10 = 0 + a(4)

∴ a = 2.5 rad/s^2.

Torque = Ia = 5*2.5 = 12.5Nm,

where ‘I’ is the moment of inertia.

Correct choice is (b) 1.8W

Easy explanation: Work done = Tθ, where T is torque & θ is angular displacement.T = r*F = 0.1*5 = 0.5Nm

& θ = 30° = π/6 rad.

∴ Work = 0.5*π/6 = π/12 J.

Moment of INERTIA about SPINNING AXIS = MR^2/2 = 2*0.01/2 = 0.01kgm^2.

Using, T = Ia, where ‘I’ is moment of inertia & ‘a’ is angular acceleration, we get:

a = T/I

∴ a = 0.5/0.01 = 50 rad/s^2.

Now, θ = w0t + (1/2)at^2 to calculate the time for 30° rotation, where wo is initial angular velocity.

∴ π/6 = 0 + 0.5*50*t^2.

∴ t = √(π/150) = 0.145 s.

∴ POWER = Work/time = (π/12) / (0.145) = 1.8 W.

Right answer is (d) Angular velocity

Explanation: Moment of inertia is the summation of PRODUCT of mass and PERPENDICULAR distance from the axis squared of each particle. More the mass, more will be its value. It depends on perpendicular distance hence, it will DEPEND on orientation and distance of particles from the axis. But, it doesn’t depend on the angular velocity. As the body having a particular value of moment of inertia can achieve any value of angular velocity depending on the torque APPLIED on it.

The correct choice is (C) a

Best EXPLANATION: The acceleration of the CENTRE of mass = (M1*a1 + m2*a2)/(m1 + m2)

m1 = m2 = m

a1 = 0

a2 = 2a

The acceleration of the centre of mass = (m*0 + m*2a) / (m + m)

 = 2ma/2m

 = a.

Right answer is (a) True

The best EXPLANATION: The centre of mass of the body always lies on the line of symmetry for an OBJECT with UNIFORM density if there exists a line of symmetry for that geometry. Because, by definition, symmetry implies an equivalence of mass on either side of the symmetrical AXIS if the density is uniform.

Correct answer is (b) 1.5 KG

The explanation is: SUM of masses = 1 + 1.5 + 2 + M = 4.5 + M

x-coordinate;

(1*2 + 1.5*1 + 2*2 – M)/(4.5 + M) = 1

4.5 + M = 7.5 – M

2M = 3

M = 1.5 kg.

The correct OPTION is (a) 4/3, 2/3, 2/3

The EXPLANATION is: Sum of masses = 2+1 = 3

x-coordinate;

(2*1 + 1*2)/3 = 4/3

y-coordinate;

(2*0 + 1*2)/3 = 2/3

z-coordinate;

(2*1 + 1*0)/3 = 2/3.

Correct option is (C) -6.67 m/s

Explanation: let the mass of part 1 be 2X, & the mass of part 2 be 3x SINCE the masses are in the ratio 2:3.2x + 3x = 10g.

∴ x = 2g. Therefore, masses of part 1 & 2 are 4g & 6g respectively.

The total momentum will be CONSERVED because there is no EXTERNAL force.Let the velocity of 2nd part be ‘v’.

∴ 0 = 4*10 + 6*v∴ v = -40/6 = -6.67 m/s.

Right option is (a) 0.48 X 10^-10 m

To explain I would say: The LOCATION of the centre of mass from an object in a two-object system is inversely proportional to the mass of the object.

The atomic mass of oxygen = 16

The atomic mass of carbon = 12

Total atomic mass = 16 + 12 = 28

The location of centre of mass relative to oxygen atom = (1.13 x 10^-10) x 12 / 28

 = 0.48 x 10^-10 m.

The correct answer is (C) 2πR

The BEST I can explain: Let the time taken for one rotation be ‘T’. Let the angular SPEED of the wheel be ‘w’.w = 2π/T.

The VELOCITY of the centre is Rw = R*2π/T.

∴ DISTANCE moved by it = R*2π/T*T

= 2πR.

Correct choice is (b) > F1

For EXPLANATION I would say: Power is defined as the work done PER unit time. For the same power, the amount of ROTATION should be the same in a GIVEN time. For the th angular acceleration should be the same in both the cases & therefore the torque should be the same.

Thus,F2* I/2 = F1* I.

F2 = F1*2 OR F2 > F1

The correct option is (B) 10cm

Easiest EXPLANATION: The angular momentum of a particle along the axis is GIVEN by Iω.I = mr^2 ω = v/R, where ‘r’ is the radius of the particle.Iω = mvr = 5∴ 5*10*r = 5∴ r = 0.1m = 10cm.

The correct answer is (a) Cross product is COMMUTATIVE

To explain: Cross product a X b ≠ b X a, therefore it is not commutative.a X (b + c) = (a X b) + (a X c), therefore it is DISTRIBUTIVE over addition.Dot product is also known as SCALAR product & a.b = b.a, so it is commutative.

The correct ANSWER is (a) 5.98m

Explanation: There is no NET force on the particle in the x-direction, so the x-coordinate of centre of mass will be equal to the range of projectile motion.The parts will have masses 2kg and 1KG.The time of motion ‘t’ = 2uy/g = 1s.The range = uxt = 5√3*1 = 8.66m.The centre of mass will land at 8.66m.So,

(m1x1 + m2x2)/(m1 + m2) = 8.66.

m1 = 2kg, m2 = 1kg, x1 = 10m, we have to calculate x2.

∴ m2x2 = (8.66*3) – 2*10

= 25.98 – 20 = 5.98.

∴ x2= 5.98/1 = 5.98m.

The CORRECT option is (b) False

To EXPLAIN: The DIAGRAM below SHOWS that the angular momentum of the particle is not along the direction of angular velocity when calculated about point A. Both the vectors are parallel when the angular momentum is calculated about point O.

The correct choice is (b) 4kgm^2

Easy explanation: Using parallel AXIS theorem we can FIND I about centre of mass.

∴ I = 4 – 2*1 = 2kgm^2.

Now using perpendicular axis theorem we GET I1 about the desired axis.

∴ I1 = I + I = 4kgm^2.

Right choice is (b) False

The best I can explain: For momentum to be CONSERVED its value should remain constant, but in projectile motion the force of gravity is always changing its momentum. The INTERNAL forces will break the PARTICLE and the momentum of different PARTICLES can be conserved in a direction perpendicular to direction of gravity.

Right OPTION is (B) I1 = I2 – ma^2

To elaborate: PARALLEL axis theorem can be used for any BODY. The parallel axis theorem states that moment of inertia about an axis perpendicular to an axis passing through centre of mass is given by:

I = ICOM + ma^2, where m is mass of the body & ‘a’ is the distance between the axes. So, I2= I1 + ma^2 OR I1 = I2 – ma^2.

Right option is (b) False

The explanation: If a rigid body has zero kinetic ENERGY means that vector sum of all forces will be zero and the TORQUE on it will be zero. For torque to be zero the vector CROSS product of force and distance should be zero. It is not necessary for all forces to pass through the centre.

The CORRECT answer is (d) 1/4, 3/4

The best I can explain: Sum of masses = 5 + 15 = 20

x-coordinate;

(5*1 + 15*0)/20 = 1/4

y-coordinate;

(5*0 + 15*1)/20 = 3/4.

Correct option is (b) 2/3 s

Easy explanation: Friction force ‘f’ will cause its speed to come down & increase its angular speed till V=Rw.

f = μN = 0.3*2*10 = 6N.

Thus, deceleration ‘a’ = 6/2 = 3m/s^2.

And, angular ACCELERATION = a/R = 3/0.05 = 60 rad/s^2.

v = V0 – at = 4 – 3T.

w = w0 + (a/R)t = 0 + 60t = 60t.

At PURE rolling, v=Rw

⇒ 4 – 3t = 0.05(60t)

∴ 4 = 3t + 3t = 6t.

∴ t = 2/3 s.

Right option is (b) 25/3π N

The best I can explain: Let w be the angular velocity & a be the angular acceleration.

Moment of inertia of the RING = MR^2

= 2*0.25 = 0.5kgm^2.

Angular distance = 3 ROTATIONS = 3*2π rad = 6π rad.

Using, w^2 = w0^2 + 2aθ, we get:

0 = 100 – 2a*6π.

a = 100/12π.

Torque = IA = rF

∴ F = Ia/r = 0.5*100/(12π*0.5) = 25/3π N.

The CORRECT answer is (a) -20i + 12j + k

To ELABORATE: The CROSS product of a & b = i(0-20) – j(0 – 12) + k(10 – 9)

= -20i + 12j + k.

Note that this is not the same as b X a.

Right option is (a) The above said situation is not possible

For explanation: When the balls COLLIDE, net external force on them is zero, so we can conserve momentum.2*10 = 3*v∴ v = 20/3 = 6.67m/s.

Note that the final kinetic ENERGY is less than the initial kinetic energy, this is because the internal forces have DONE work during collision which gets converted into SOUND & heat energy.

The correct choice is (b) may move

To EXPLAIN: Since the vector ADDITION of all EXTERNAL forces is zero, we know that the acceleration vector is zero for the resultant force vector as mass cannot be zero. Zero acceleration can IMPLY zero or non-zero constant velocity. Thus, the CENTRE of mass may move.

Correct ANSWER is (a) 1/4 (-16i + 17j + 7k) m

For explanation: SUM of masses = 1 + 3 = 4

x-coordinate;

(1*2-3*6)/4 = -16/4

y-coordinate;

(1*5 + 3*4)/4 = 17/4

z-coordinate;

(1*13-3*2)/4 = 7/4

Position of CENTRE of MASS = 1/4 (-6i + 17j + 5k) m.

Right choice is (a) 0.25 s

Easy explanation: Friction force will DECREASE its linear velocity & increase its ANGULAR velocity.

F = Ma .Friction ‘f’ = μN = 0.4*1*10 = 4N.

Thus, v = v0 – at&w = w0 + (a/R)t.

For pure rolling v = wR,

thus: v0 – at = R(w0 + (a/R)t).a = f/M∴ 5 – 4/1 = 0.1(0 + (4/0.1)t)

∴ 10 = 40t

OR t = 0.25 s.

The CORRECT choice is (b) 3:1

Easiest explanation: The kinetic energy of a pure rolling disc as seen from the ground frame is GIVEN by:1/2MV^2 + 1/2Iw^2.

I = MR^2/2w = V/R

∴ 1/2MV^2 + 1/2Iw^2 = 1/2MV^2 + 1/4MV^2 = 3/4MV^2.

Kinetic energy as seen from the MOVING frame will only be the energy DUE to rotation about the disc’s axis.

∴ Kinetic energy from moving frame = 1/2Iw^2 = 1/4MV^2.

∴ Ratio = (3/4MV^2) / (1/4MV^2)

= 3:1.

Correct option is (b) 9.09 rad/s

The best EXPLANATION: The FRICTION acting between the two SURFACES will ensure that the particle stays on the DISC and doesn’t fly out. The angular momentum of the system will be conserved because there is no external torque.Moment of INERTIA of disc = MR^2/2 = 2*0.01/2 = 0.01kgm^2.

Moment of inertia with particle on disc = 0.01 + mR^2

= 0.01+0.1*0.01 = 0.011kgm^2.

∴ I1w1 = I2w2

∴ 0.01*10 = 0.011*w2

∴ w2 = 9.09 rad/s.

The correct answer is (a) 0

The explanation: Since both A and B are moving towards each other due to mutual force of attraction, we can INFER that the particles of the system are in motion due to internal forces and hence the state of CENTRE of MASS will remain unchanged, i.e., at REST.

Correct answer is (b) False

Explanation: The angular MOMENTUM of a PARTICLE rotating about a fixed axis is given by L = r X p. At any time the vector p is fixed. But from different points on the axis, we will GET different r VECTORS. So, the cross product of r & p will be different. Hence, the angular momentum vector is different about different points on the axis.

The correct choice is (a) (i + 2j + 3K) m/s

To elaborate: The VELOCITY of the centre of mass = (m1*v1 + M2*v2 + m3*v3)/(m1 + m2 + m3)

 = (10*10i + 20*10j + 30*10k) / (10 + 20 + 30)

 = (100i + 200j + 300k) / 100

 = (i + 2j + 3k) m/s.

Correct option is (c) 1.25s

For explanation: Let R be the radius, I be the MOMENT of inertia, a be the angular ACCELERATION & F be the FORCE applied.Moment of inertia of disc about given axis = MR^2/2 = 0.5 kgm^2.Torque = r X F = 2Nm.

But, torque is also = Ia.∴ a = 2/0.5 = 4 rads^2.Using, w = w0 + a*DT, we get:0 = 5 – 4dt, where dt is the time interval.

∴ dt = 5/4 = 1.25s.

Right CHOICE is (c) Same for both

The explanation is: The ANGULAR momentum for each will be the same as torque is the same.Let the mass of the ring be m, and their radius be ‘R’. The ratio of angular speeds will be the inverse ratio of their moments of INERTIA.Moment of inertia of ring = mR^2& moment of inertia of DISC = 2mR^2/2 = mR^2. As their moment of inertia is the same their angular speeds will be the same.

Right answer is (c) 0.25Nm

The explanation is: The force F1 PASSES through the axis of ROTATION, so it will not PRODUCE a torque.

The force F2 is perpendicular to axis and RADIUS, so it will provide a torque = r*F2 about the axis of rotation, where ‘r’ is the distance of point of application of force F2 from the axis of rotation.

Therefore, torque = r* F2= 0.05 * 5 NM

= 0.25Nm.

Correct option is (B) Linear momentum may be conserved

For EXPLANATION I would say: The TWO FORCES acting on the body may or may not have a net vector sum of zero, depending on which linear momentum may or may not be conserved. Thus, the option ‘linear momentum may be conserved’ is correct.

Correct choice is (a) 0m/s

The best explanation: Initially the boy was at rest and there is no friction, so the net force on the ball-boy SYSTEM is zero.Hence, their momentum change will be zero and FINAL VELOCITY of centre will also be zero.Note that boy will move BACKWARDS with a speed of 2*10/50 = 0.4m/s.

Right answer is (b) False

The BEST explanation: On collision of two balls, there is no EXTERNAL FORCE, so the total MOMENTUM will be conserved. But due to internal forces there could be loss of energy in the form of heat, sound or deformation (potential energy) and hence final kinetic energy will be less than or equal to initial kinetic energy.

The correct option is (a) 6 m from 4 kg BODY

For explanation: The two bodies would meet at their centre of mass.

Sum of MASSES = 2 + 3 = 5 kg

Let us consider the 2 kg body as a REFERENCE;

The centre of mass from the 2 kg body =(3 X 10) / 5

 = 6 m.

Correct option is (b) False

The best explanation: The CENTRE of MASS for an OBJECT need not NECESSARILY lie inside the object. A ring or a C-section element are classic EXAMPLES for this.

Correct ANSWER is (C) π/75 rad/s^2

Easiest explanation: Initial ANGULAR speed = 2rpm = π/15 rad/s.Final angular speed = 2π/15 rad/s.

Using, w = W0 + a*dt, we get:

2π/15 = π/15 + 5a.

∴ π/15 = 5*a

OR a = π/75 rad/s^2.

Correct choice is (a) 0 cm

Easiest explanation: SINCE the plate is FREE, the net force on the plate is ZERO. So, there will be no shift in the CENTRE of mass of the plate, i.e., the shift in the centre of mass of the plate will be 0 cm.

Right option is (B) LESSER than or equal to D

Explanation: Since all objects are placed at the same distance, they are either placed in a spherical CONFIGURATION, circular-arc configuration or a LINEAR configuration. In all of the cases, the centre of mass of the system will lie within the GEOMETRY of the shape. Hence the centre of mass of the system is at a position lesser than or equal to D.