JEE Physics Centre of Mass MCQs Set 02

Physics Objective Questions and Answers: Centre of Mass

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Multiple Choice Questions

Question 1: The motion of the centre of mass of a two-particle system is unaffected by their internal forces:
(a) only if they act along the line joining the particles
(b) only if they act at right angles to the line joining the particles
(c) only if they are obliquely inclined to the line
(d) irrespective of the directions of the internal forces
Show Answer & Explanation

Answer: (d) irrespective of the directions of the internal forces

Explanation:
1. Newton's third law gives equal and opposite internal forces
2. Their vector sum is zero for any direction

Teacher's Note:
1. Direction of internal forces does not matter for the centre of mass.

Question 2: A ball in a closed box makes collisions with the walls while the box rests on a smooth surface. The centre of mass:
(a) of the box remains constant
(b) of the box plus ball system remains constant
(c) of the ball remains constant
(d) of the ball relative to the box remains constant
Show Answer & Explanation

Answer: (b) of the box plus ball system remains constant

Explanation:
1. Net external horizontal force on box plus ball is zero
2. So only the combined centre of mass stays in place

Teacher's Note:
1. Each part moves, but the combined centre of mass cannot.

Question 3: A man of 60 kg walks 4 m along a boat of 120 kg floating on still water (no friction). The displacement of the boat relative to the water is:
(a) \( 4 \) m
(b) \( \frac43 \) m
(c) \( 2 \) m
(d) \( \frac83 \) m
Show Answer & Explanation

Answer: (b) \( \frac43 \) m

Explanation:
1. Let boat move x opposite to the man; man moves \( 4-x \)
2. \( 60(4-x)=120x \)
3. \( 240=180x\Rightarrow x=\frac43 \) m

Teacher's Note:
1. Relative displacement 4 m is shared in the inverse ratio of the masses.

Question 4: A block A (mass 4M) rests on a smooth wedge B (mass 20M, base length l) on a smooth floor. When the block slides to the lowest end of the wedge, the wedge has moved a distance:
(a) \( \frac l5 \)
(b) \( \frac l4 \)
(c) \( \frac l6 \)
(d) \( \frac l3 \)
Show Answer & Explanation

Answer: (c) \( \frac l6 \)

Explanation:
1. Block moves \( l-x \) horizontally relative to the ground, the wedge moves x opposite
2. \( 4M(l-x)=20Mx \)
3. \( l=6x\Rightarrow x=\frac l6 \)

Teacher's Note:
1. Horizontal centre of mass stays fixed since there is no external horizontal force.

Question 5: An isolated particle of mass m moves along the x-axis in the xy-plane. It suddenly explodes into two pieces of masses m/4 and 3m/4. A moment later the lighter fragment is at y = +15 cm. The y-coordinate of the heavier fragment is:
(a) +5 cm
(b) -15 cm
(c) -5 cm
(d) -3.75 cm
Show Answer & Explanation

Answer: (c) -5 cm

Explanation:
1. The centre of mass remains on the x-axis (y = 0)
2. \( 0=\frac m4(15)+\frac{3m}4y \)
3. \( y=-5 \) cm

Teacher's Note:
1. Use the fact that the original motion was along y = 0.

Question 6: A block of mass m is released at one end inside a smooth thin semicircular tube of the same mass m and mean radius R, the tube standing in a vertical plane on a smooth floor. By the time the block reaches the other end of the tube, the displacement of the tube is:
(a) \( \frac R2 \)
(b) \( \frac{2R}{\pi} \)
(c) \( \frac{4R}{\pi} \)
(d) \( R \)
Show Answer & Explanation

Answer: (d) \( R \)

Explanation:
1. The block moves a horizontal distance 2R relative to the tube
2. Let the tube move x opposite: block moves \( 2R-x \) relative to the ground
3. \( mx=m(2R-x)\Rightarrow x=R \)

Teacher's Note:
1. The two ends of the tube are 2R apart horizontally, not R.

Question 7: A man of mass m climbs a rope suspended from a balloon of mass M. The balloon is initially at rest. If the man climbs at speed v relative to the rope, the balloon moves:
(a) upward with speed \( \frac{Mv}{m+M} \)
(b) downward with speed \( \frac{mv}{M} \)
(c) downward with speed \( \frac{(M+m)v}{M} \)
(d) downward with speed \( \frac{mv}{m+M} \)
Show Answer & Explanation

Answer: (d) downward with speed \( \frac{mv}{m+M} \)

Explanation:
1. Let the balloon move down at \( V_b \); the man moves up at \( v-V_b \) relative to the ground
2. \( m(v-V_b)-MV_b=0 \)
3. \( V_b=\frac{mv}{m+M} \)

Teacher's Note:
1. Relative speed is the sum of the two ground speeds.

Question 8: A shell moves along a parabolic path and explodes at the highest point into two equal pieces. One piece falls vertically downward from rest. The other piece:
(a) also falls vertically
(b) moves forward with the same speed
(c) moves backward
(d) moves forward with twice the speed of the shell at that point
Show Answer & Explanation

Answer: (d) moves forward with twice the speed of the shell at that point

Explanation:
1. At the top the velocity is horizontal, \( u \)
2. \( mu=\frac m2(0)+\frac m2V \)
3. \( V=2u \)

Teacher's Note:
1. Use momentum conservation in the horizontal direction.

Question 9: A shell fired at 100 m/s at 60 degrees to the horizontal explodes at the top of its path into two equal parts. One part moves backward along the original horizontal line at 50 m/s. The speed of the other part is:
(a) 100 m/s
(b) 50 m/s
(c) 150 m/s
(d) 200 m/s
Show Answer & Explanation

Answer: (c) 150 m/s

Explanation:
1. Speed at the top \( =100\cos60^\circ=50 \) m/s
2. \( 2m(50)=m(-50)+mV \)
3. \( V=150 \) m/s

Teacher's Note:
1. Use the horizontal component only; the vertical component is zero at the top.

Question 10: A shell at rest at the origin explodes into three fragments of masses 1 kg, 2 kg and m kg. The 1 kg and 2 kg pieces fly off at 12 m/s along the x-axis and 8 m/s along the y-axis respectively. If the third piece has speed 40 m/s, the total mass of the shell is:
(a) 3.5 kg
(b) 3.0 kg
(c) 4.0 kg
(d) 2.5 kg
Show Answer & Explanation

Answer: (a) 3.5 kg

Explanation:
1. Momentum of the first two: \( 12\hat i+16\hat j \), magnitude 20 kg m/s
2. \( m(40)=20\Rightarrow m=0.5 \) kg
3. Total = 1 + 2 + 0.5 = 3.5 kg

Teacher's Note:
1. The third fragment must cancel the vector sum of the other two momenta.

Question 11: A block moving at 20 m/s on a smooth surface bursts into two equal parts which continue in the same direction. If one part moves at 30 m/s, the speed of the other part and the fractional change in the kinetic energy of the system are:
(a) 10 m/s and an increase by \( \frac14 \)
(b) 10 m/s and a decrease by \( \frac14 \)
(c) 25 m/s and an increase by \( \frac14 \)
(d) 15 m/s and an increase by \( \frac12 \)
Show Answer & Explanation

Answer: (a) 10 m/s and an increase by \( \frac14 \)

Explanation:
1. \( m(20)=\frac m2V+\frac m2(30)\Rightarrow V=10 \) m/s
2. \( K_i=200m \)
3. \( K_f=\frac12\frac m2(100)+\frac12\frac m2(900)=250m \)
4. \( \frac{250m-200m}{200m}=\frac14 \) increase

Teacher's Note:
1. Momentum is conserved; kinetic energy rises due to the explosion's internal energy.

Question 12: Blocks A (40 kg) and B (60 kg) on a smooth surface are joined by a spring of constant 600 N/m that is stretched by 1.5 m. They are released from rest. The speeds of the blocks when the spring becomes unstretched are:
(a) 3 m/s and 4.5 m/s
(b) 4.5 m/s and 3 m/s
(c) 6 m/s and 4 m/s
(d) 2.25 m/s and 1.5 m/s
Show Answer & Explanation

Answer: (b) 4.5 m/s and 3 m/s

Explanation:
1. Momentum: \( 40V_1=60V_2\Rightarrow 2V_1=3V_2 \)
2. Energy: \( \frac12(600)(1.5)^2=675=\frac12(40)V_1^2+\frac12(60)V_2^2 \)
3. Substituting \( V_2=\frac23V_1 \): \( 675=\frac{100}{3}V_1^2 \), so \( V_1=4.5 \) m/s and \( V_2=3 \) m/s

Teacher's Note:
1. The lighter block gets the larger speed.

Question 13: A stationary body explodes into two fragments of masses m1 and m2. If the momentum of one fragment is p, the energy of the explosion is:
(a) \( \frac{p^2}{2(m_1+m_2)} \)
(b) \( \frac{p^2}{2\sqrt{m_1m_2}} \)
(c) \( \frac{p^2}{2(m_1-m_2)} \)
(d) \( \frac{p^2(m_1+m_2)}{2m_1m_2} \)
Show Answer & Explanation

Answer: (d) \( \frac{p^2(m_1+m_2)}{2m_1m_2} \)

Explanation:
1. Both fragments have momentum magnitude p
2. \( E=\frac{p^2}{2m_1}+\frac{p^2}{2m_2} \)
3. \( =\frac{p^2(m_1+m_2)}{2m_1m_2} \)

Teacher's Note:
1. The energy equals the total kinetic energy of the fragments.

Question 14: A 5 kg gun fires a 50 g bullet with a speed of 400 m/s (relative to the ground). The recoil speed of the gun is:
(a) 40 m/s
(b) 4 m/s
(c) 0.4 m/s
(d) 20 m/s
Show Answer & Explanation

Answer: (b) 4 m/s

Explanation:
1. \( 5V=0.05\times400 \)
2. \( V=4 \) m/s

Teacher's Note:
1. Momentum before firing is zero.

Question 15: A block moving in air explodes into two parts. Just after the explosion (neglecting gravity during the burst):
(a) the total momentum of the two parts equals the momentum of the block before the explosion
(b) the total kinetic energy of the parts equals that of the block
(c) the total momentum must change
(d) the total kinetic energy must decrease
Show Answer & Explanation

Answer: (a) the total momentum of the two parts equals the momentum of the block before the explosion

Explanation:
1. Internal forces cannot change total momentum
2. Kinetic energy increases because stored chemical energy is released

Teacher's Note:
1. Momentum is conserved in all explosions.

Question 16: A bomb dropped from an aeroplane explodes in air. Its total:
(a) momentum decreases
(b) momentum increases
(c) kinetic energy increases
(d) kinetic energy decreases
Show Answer & Explanation

Answer: (c) kinetic energy increases

Explanation:
1. Explosion forces are internal, so momentum is unchanged
2. Internal energy converts into kinetic energy

Teacher's Note:
1. Energy is added from the explosive's stored energy.

Question 17: Two particles A and B, initially at rest, move toward each other under their mutual attraction. At the instant the speed of A is v and that of B is 2v, the speed of the centre of mass is:
(a) v
(b) 2v
(c) zero
(d) 3v/2
Show Answer & Explanation

Answer: (c) zero

Explanation:
1. The net external force is zero
2. The initial momentum is zero, so the centre of mass stays at rest

Teacher's Note:
1. Mutual forces are internal.

Question 18: Two particles approach each other. After the collision, one of the particles has momentum p in the centre of mass frame. In the same frame, the momentum of the other particle is:
(a) zero
(b) \( -\frac{\vec p}2 \)
(c) \( -\vec p \)
(d) \( -2\vec p \)
Show Answer & Explanation

Answer: (c) \( -\vec p \)

Explanation:
1. Total momentum in the centre of mass frame is zero
2. \( \vec p+\vec p'=0\Rightarrow\vec p'=-\vec p \)

Teacher's Note:
1. The centre of mass frame has zero total momentum by definition.

Question 19: A train of mass M moves with constant speed V on a circular track of radius R. The length of the train is half the perimeter of the track. The linear momentum of the train is:
(a) zero
(b) \( \frac{2MV}\pi \)
(c) \( MVR \)
(d) \( MV \)
Show Answer & Explanation

Answer: (b) \( \frac{2MV}\pi \)

Explanation:
1. Treat the train as a semicircular ring of mass M; its centre of mass is \( \frac{2R}\pi \) from the centre
2. \( v_{cm}=\frac{2R}{\pi}\omega=\frac{2R}\pi\frac VR=\frac{2V}\pi \)
3. \( p=Mv_{cm}=\frac{2MV}\pi \)

Teacher's Note:
1. Momentum of a system is M times the velocity of its centre of mass.

Question 20: If the kinetic energy of a body becomes four times its initial value, its momentum increases by:
(a) 50%
(b) 125%
(c) 150%
(d) 100%
Show Answer & Explanation

Answer: (d) 100%

Explanation:
1. \( K\propto p^2 \) (fixed mass)
2. \( \frac{p_2}{p_1}=\sqrt4=2 \)
3. Increase \( =(2-1)\times100=100\% \)

Teacher's Note:
1. Kinetic energy varies as the square of momentum.

Question 21: Two bodies of masses m and 4m have equal linear momentum. The ratio of their kinetic energies is:
(a) 1 : 4
(b) 1 : 1
(c) 4 : 1
(d) 1 : 2
Show Answer & Explanation

Answer: (c) 4 : 1

Explanation:
1. \( K=\frac{p^2}{2m} \)
2. \( \frac{K_1}{K_2}=\frac{4m}{m}=4 \)

Teacher's Note:
1. With equal momentum the lighter body has more energy.

Question 22: Two observers in different inertial frames measure the momentum of a body. Then:
(a) the measured momenta may be equal
(b) the momenta must be equal
(c) the kinetic energies must be equal
(d) none of the above is possible
Show Answer & Explanation

Answer: (a) the measured momenta may be equal

Explanation:
1. \( \vec p=m\vec v \) depends on the frame's velocity
2. The two values are equal only if the frames have the same velocity

Teacher's Note:
1. Velocity is frame dependent, so momentum is also.

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