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Question 1

CM of the sector ( Y cm )

Answer 1

Y CM = 2r/3 [(sin θ/2 ) / θ/2 ]

Question 2

When the linear mass density of a non-uniform rod is given in terms of alpha + beta.x

Lamda = alpha (a) + beta.x (b)

Then it’s CM neglecting it’s thickness and it’s length is L

Answer 2

X CM = (3aL + 2bL^2) / 3( 2a + bL)

Question 3

CM of uniform triangle from head to base

And base to head

Answer 3

Head to base : ( 2L/3, 2L/3 )

Base to head : ( L/3, L/3 )

Question 4

CM of uniform arc

And for along the bisector

Answer 4

Y CM = R / θ [2.sin ( θ/2 ) ]
Or = 2R/ θ [ sin (θ/2) ]

For along the bisector: R[ (sin θ/2 ) /( θ/2) ]

Question 5

CM of the semi-circular ring ( Y cm )

Answer 5

Y CM = 2R/π

Question 6

CM of Uniform semi-circular disc

Answer 6

Distance of CM from base = 4r/3π

Question 7

CM of hollow semi-spherical shell

And solid hemisphere

Answer 7

Semi: R/2

Solid: 3r/8

Question 8

CM for hollow cone

And

CM for solid cone

Answer 8

Hollow: H/3 from base

Solid: H/4 from base

Question 9

Area of cap

Answer 9

Area= 2π R^2 ( 1- cos θ )

Question 10

Volume of paraboloid

Answer 10

Vol = π/2.h.R^2 from base

Question 11

CM of paraboloid

Answer 11

2H/3 from the top point ( from the tip point )

H/3 from the base ( from the bottom circular part )

Question 12

CM of equilateral triangle from base

Answer 12

CM : L/ 2√3

Question 13

Two blocks of mass m1 & m2 are connected with a spring. One of the block is being pulled with a velocity of V° then Xmax =?

Answer 13

When it will be pulled then a point will come when both the blocks will be moving with the same velocity let that velocity be v so,

X max or X min = V° √ [ m1m2 / ( m1 + m2 ) k ]

Question 14

In case of collision: two bodies are moving in the same direction ( magnitude of one being greater than the other ) and they collide and then they start moving with the same velocity then loss in K.E. =?

Answer 14

Loss in K.E.= 1/2 ( μ. V relative ^2)

μ = m1m2 / m1+m2

Question 15

2 bodies one moving with u1 & other with u2 & after the collision 1 moving with V1 & other with V2 & coefficient of restitution is e then V1 & V2 =?

Answer 15

V1 = [ m2 / m1+ m2 ] (1+ e) .u2 + [ m1 - em2 / m1 + m2] .u1

V2 = [ m1 / m1+ m2 ] (1+ e) .u1 + [ m2 - em1 / m1 + m2] .u2

Question 16

2 bodies moving in the same direction one is very very big in comparison to the other and it is moving with a velocity of V° & smaller one is moving with a velocity of U° Then when the bigger body collides with the smaller one the velocity of the smaller one and the bigger one will be?

Answer 16

V2 ( velocity of smaller one ) =2V° - U°

V1 ( bigger one ) velocity will remain same

In case they will moving in direction of each other then

V2 = 2V° + U°

Question 17

M1 & M2 are moving in the same direction with a velocity of u1 & u2 after collision both moves with velocity of v1 & v2 then heat loss =?

Answer 17

Heat loss = 1/2 μ ( 1 - e^2 ) Urel ^2

Question 18

For a variable mass system when F external = 0 & F external Is not equal to 0 then F* =?

Answer 18

F* for F external =0
F*= μVrel ( opposite of Vrel )

F* for F external not equal to 0 then:
F° + F* = M.dv/dt