Center of Mass MCQ Quiz in தமிழ் - Objective Question with Answer for Center of Mass - இலவச PDF ஐப் பதிவிறக்கவும்

CONCEPT:

Centre of mass:

• The centre of mass of a body or system of a particle is defined as, a point at which the whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.

The motion of the centre of mass:

• Let there are n particles of masses m1, m2,..., mn.
• If all the masses are moving then,

⇒ Mv = m1v1 + m2v2 + ... + mnvn

⇒ Ma = m1a1 + m2a2 + ... + mnan

⇒ 

⇒ M = m1 + m2 + ... + mn

• Thus, the total mass of a system of particles times the acceleration of its centre of mass is the vector sum of all the forces acting on the system of particles.
• The internal forces contribute nothing to the motion of the centre of mass.

CALCULATION:

Given: , and v = 0 m/sec

Where v = speed of the centre of mass

• Since both the cars are moving in opposite directions therefore if we take the velocity of car A as positive then the velocity of car B will be negative.

Let v_A = velocity of car A and vB = velocity of car B

If v is the velocity of the centre of mass, then,

⇒ Mv = mAvA - mBvB

⇒ 0 = mAvA - mBvB

⇒ mAvA = mBvB

⇒ 

⇒ 

• Hence, option 2 is correct.

The correct answer is option 4) i.e. none of the above

CONCEPT:

• Center of mass: Center of the mass of a body is the weighted average position of all the parts of the body with respect to mass.
  • The center of mass is used in representing irregular objects as point masses for ease of calculation.
  • For simple-shaped objects, its center of mass lies at the centroid.
  • For irregular shapes, the center of mass is found by the vector addition of the weighted position vectors.
• The position coordinates for the center of mass can be found by: 
     

EXPLANATION:

• For the centre of mass to be at the origin, the sum of the product of the mass and respective distances from the origin must equal to zero.
• That means the centre of mass depends on the mass and distance simultaneously.
• The first three options only indicate a relationship with the number of particles and mass separately.
• Hence, none of the above options relates to the given statement.

CONCEPT:

Centre of mass:

• The centre of mass of a body or system of a particle is defined as, a point at which the whole of the mass of the body or all the masses of a system of particle appeared to be concentrated.

The motion of the centre of mass:

• Let there are n particles of masses m1, m2,..., mn.
• If all the masses are moving then,

⇒ Mv = m1v1 + m2v2 + ... + mnvn

⇒ Ma = m1a1 + m2a2 + ... + mnan

⇒ 

⇒ M = m1 + m2 + ... + mn

• Thus, the total mass of a system of particles times the acceleration of its centre of mass is the vector sum of all the forces acting on the system of particles.
• The internal forces contribute nothing to the motion of the centre of mass.

EXPLANATION:

• In the given question a body is projected at an angle with some initial velocity from the surface of the earth and follows the parabolic path.
• The body explodes somewhere during the flight and splits into a number of particles.
• When the body explodes only internal forces acts on the body and we know that the internal forces contribute nothing to the motion of the centre of mass.
• Therefore the centre of mass will follow the same parabolic path after the explosion. Hence, option 3 is correct.

The correct answer is option 3) i.e. positive for some particles and negative for other particles.

CONCEPT:

• Centre of mass: It is the point of a body where the whole mass of the body is assumed to be concentrated.
  • The Centre of mass is also the point about which each particle of a body is symmetrically distributed on all sides.

EXPLANATION:

• If the centre of mass is at the origin, this means that some particles will be towards the left of the COM and some towards the right of the COM.
• The particles towards the left of COM will have a negative value of X-coordinate whereas those towards the right will have a positive value of X-coordinate.
• In this case, the particles can not have always positive values of X-coordinates or always negative values of X-coordinates separately because the particles are uniformly distributed about the centre of mass of the body.

Explanation:

Centre of mass (COM): Centre of mass can be described as an imaginary point, where the whole mass of the system can be supposed to be concentrated. Also, the motion of the system can be defined in terms of centre of mass.

The centre of mass (C.O.M) of the body does not necessarily lie inside the solid body, It may be inside or outside of the solid body.

C.O.M of some symmetrical body can be given as

• The centre of mass may lie outside the body like the ring, hollow sphere, hollow cylinder etc.
• Also, the centre of mass will lie at its geometrical axis for bodies like a solid sphere, disc, solid cylinder etc.

Hence the correct option is not need to lie within the body.

CONCEPT:

Center of mass (RCM): 

• The center of mass of the body is a point where the whole mass of the body may be assumed to be concentrated for describing its translatory motion.
• The center of mass of a system of particles is that single point that moves in the same way in which a single particle having the total mass of the system and acted upon by the same external force would move.

Mathematically center of mass for n – particle is written as

EXPLANATION:

Given -  m1 = 2 kg, m2 = 4 kg and m₃ = 4 kg

m₁ is at (1, 0, 0)

m₂ is at (1, 1, 0)

m₃ is at (0, 1, 0)

• The Centre of mass in direction of the x−axis is 

• The Centre of mass in direction of the y−axis is 

The position vector of the center of mass will be,

r_cm =  

So, the correct answer is option (1)

The correct answer is option 1) i.e. The center of mass of a system of two particles lie on the lines joining the particles.

CONCEPT:

• Center of mass: Center of the mass of a body is the weighted average position of all the parts of the body with respect to mass.
  • The center of mass is used in representing irregular objects as point masses for ease of calculation.
  • For simple-shaped objects, its center of mass lies at the centroid.
  • For irregular shapes, the center of mass is found by the vector addition of the weighted position vectors.
• The position coordinates for the center of mass can be found by: 
     

EXPLANATION: 

• Using the equation for position vector, 
• Let us assume a case when the centre of mass is in origin i.e. 
  
  

This indicates that  and  are opposite to each other. Similarly, for any other point of the centre of mass, this principle can be applied.

This proves that the centre of mass of a system of two particles lies on the line joining the particles.

CONCEPT:

• Centre of mass: Centre of Mass of a body is a point at which the whole of the mass of the body appeared to be concentrated.

The Centre of masses of different bodies is given below in the table: 

EXPLANATION:

• From the above table, it is clear that the center of mass of a hollow hemisphere is at a distance of R/2 from the center.
• So the correct answer is option 3.

The correct answer is option 1) i.e. 30 cm towards right

CONCEPT:

• Center of mass: Center of the mass of a body is the weighted average position of all the parts of the body with respect to mass.
  • The centre of mass is used in representing irregular objects as point masses for ease of calculation.
  • For simple-shaped objects, its centre of mass lies at the centroid.
  • For irregular shapes, the centre of mass is found by the vector addition of the weighted position vectors.
• The position coordinates for the centre of mass can be found by: 
     

CALCULATION:

• Assuming that the boat has uniform density, the centre of mass of the boat will be located at 1 m distance from each end.
• Let point A correspond to the position of the man who weighs 80 kg and point B correspond to the position of the man who weighs 50 kg.

Given that:

Mass of the boat, M = 70 kg

Mass of the men, m_A = 80 kg and m_B = 50 kg

Distance with respect to point A,

x_A = 0, x_B = 2 m, x_M = 1 m

Using, 

When they interhchange their positions,

xA = 2, xB = 0 m, xM = 1 m

Displacement = 

Thus, on interchanging positions, the displacement is 30 cm towards the right.

Concept

Calculation: 

Taking 1 kg as the origin

xcom= (m1x1 + m2x2 + m3x3)/(m1+m2+m3)

⇒ (1 × 0 + 1.5 × 3 + 2.5 × 0)/5

xcom = 0.9

⇒ycom = (m1y1 + m2y2 + m3y3)/ (m1 + m2 + m3)

⇒ycom = (1 × 0 + 1.5 × 0 + 2.5 × 4)/5

⇒ycom = 2

Centre of mass is at (0.9, 2)

∴ The correct answer is option (1): 0.9 cm right and 2.0 cm above 1 kg mass