Center of Mass

Question 1

Express Center of Mass as a position vector r in a mulitparticle system and also in its component form?

Answer 1

r_c = (m₁r₁+m₂r₂+m₃r₃)/(m₁+m₂+m₃)
And in coordinate form
x_c = (m₁x₁+m₂x₂+m₃x₃)/(m₁+m₂+m₃),
y_c = (m₁y₁+m₂y₂+m₃y₃)/(m₁+m₂+m₃),
z_c = (m₁z₁+m₂z₂+m₃z₃)/(m₁+m₂+m₃),

Question 2

Express Center of Mass of a 2-particle system of mass m₁ and m₂?
Also state the formula for distance of CoM (m₁ and m₂) from each of the 2 particles?

Answer 2

r_c = (m₁r₁+m₂r₂)/(m₁+m₂),
And
r₁ = m₂r₂/(m₁+m₂)
r₂ = m₁r₁/(m₁+m₂)
m₁r₁=m₂r₂

Question 3

What is the general formula of CoM of a continuous mass distribution ? state for both Position Vector r_c and component vectors x_c , y_c , z_c

Answer 3

r_c = ∫r. dm / ∫ dm,
= 1/M * ∫r. dm.

In terms of component vectors,
x_c = ∫x. dm / ∫ dm,
y_c = ∫y. dm / ∫ dm,
z_c = ∫z. dm / ∫ dm,

Question 4

What is the CoM of a semi-circular ring of radius R?

Answer 4

CoM of a semi-circular ring =
2R/∏

Question 5

What is the CoM of an arc of ring of radius R?

Answer 5

CoM of an arc of ring =
RSinϴ/ϴ

where ϴ is in radians

Question 6

What is the CoM of a semi-circular Disc of radius R?

Answer 6

CoM of a semi-circular disc = 4R/3∏

Question 7

What is the CoM of a sector of Disc of radius R?

Answer 7

CoM of a sector of disc = 2R/3 Sinϴ/ϴ

Question 8

What is the CoM of a Hollow HemiSphere of radius R?

Answer 8

CoM of a Hollow HemiSphere = R/2

Question 9

What is the CoM of a Solid HemiSphere of radius R?

Answer 9

CoM of a Solid HemiSphere = 3R/8

Question 10

What is the CoM of a Hollow Cone of height h?

Answer 10

CoM of a Hollow Cone = h/3

Question 11

What is the CoM of a Solid Cone of height h?

Answer 11

CoM of a Solid Cone = h/4

Question 12

What is the CoM of a Triangular Lamina of unequal sides?

Answer 12

CoM of a Triangular Lamina of unequal sides is the Centroid of the Triangle

Question 13

What is the CoM of a body of non-uniform mass distribution

Answer 13

CoM = ∫y. dm / ∫ dm,
dm = σ_r * dA, and
y = CoM of dm

Question 14

How do you calculate CoM of combined shapes

Answer 14

Step 1:- Calculate the CoM of the shapes.
Step 2 :- Calculate the CoM of the CoMs of the shapes calculated in Step 1.

Question 15

How do you calculate CoM of a shape if mass is not given but density is given?

Answer 15

1. For a Lamina or a Disc, we calculate the mass using the raltionship between mass (m), area density (σ) and area (A) of shape.
   * Formula is σ = m /A
2. For a Solid Shape, we calculate the mass using the raltionship between mass (m), volmerho density (σ) and area (A) of shape.
   * Formula is 𝜌 = m /A

Question 16

Explain the two general methods to calculate CoM in cavity problems

Answer 16

Method 1 :-
Use the standard formula for CoM calculation but add a negative sign to the mass being taken out.
r_c = (m₁r₁+(-)m₂r₂)/(m₁+(-)m₂),

Method 2:-
Step 1 : - Calculate the remaining mass of the object.
Step 1 : - Assume the distance of the remaining mass of object to be x
Step 3 : - Using the distance, mass proportionality equation, calucate x as other variables are known
E.g , m₁x=m₂r₂

Question 17

State the formula for displacement of CoM as a position vector r⃗ in a mulitparticle system and also in its component form?

Answer 17

Δr⃗_c = (m₁Δr⃗₁+m₂Δr⃗₂+m₃Δr⃗₃)/(m₁+m₂+m₃)

Question 18

Write the expressions for

1. Velocity
2. Momentum
3. Acceleration
4. Force

of CoM

Answer 18

r⃗_c = (m₁r⃗₁+m₂r⃗₂+….)/(m₁+m₂+….)

Differentiating wrt Time, we get
dr⃗_c/dt = v⃗_cm= (m₁v⃗₁+m₂v⃗₂+….)/(m₁+m₂+….) ⇒ Mv⃗_cm = P⃗_net

Differentiating further wrt Time, we get
dv⃗_c/dt = a⃗_cm= (m₁a⃗₁+m₂a⃗₂+….)/(m₁+m₂+….) ⇒ Ma⃗_cm = F⃗_net

Question 19

Can the equation Ma⃗_cm = F⃗_net
be written as
Ma⃗_cm = F⃗_ext? Explain why?

Answer 19

Yes,
Ma⃗_cm = F⃗_net
can be written as
Ma⃗_cm = F⃗_ext.
This is because from Newton’s 3rd law of motion all internal forces by the particles cancel out each other. Only the external forces contribute to the equation.

Question 20

What is the physical explanation of the equation Ma⃗_cm = F⃗_ext?

Answer 20

Ma⃗_cm = F⃗_ext states that center of mass of a system of particle moves as if the all the mass of the system is concentrated at the center and all the external forces are applied at this point

Question 21

What will be the path of the CoM of the fragments of projectile after it explodes mid-air? And under what condition will this path be followed.

Answer 21

The CoM of the fragments of Projectile will continue along the same parabolic path which it would have followed if there were no explosion, provided BOTH the fragments hit the ground at the same time.

Question 22

What does the law of conservation of momentum state?

Answer 22

The law of conservation of momentum states that in an isolated system the total momentum of two or more bodies acting upon each other remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed.