Physics: System of Particles and Center of Mass

Question 1

position vector of com for system of N particles (for coordinates just replace r with x y or z respectively )

Accepted Answer

r=(m1r1+m2r2+....+mnrn)/(m1+m2+....+mn)

Question 2

2 point system m1 and m2(com)

Accepted Answer

xcm=m*0+(m2)r/(m1+m2) r1=(m2)r/(m1+m2) r2=(m1)r/(m1+m2)

Question 3

position of com of continuous mass system

Accepted Answer

r =integral (dm *r)/integral dm

Question 4

com for various shapes (uniform rod, square lamina, semicircular ring, hemispherical shell ,solid circular cone ,solid hemisphere respectively)

Accepted Answer

(L/2,0) (L/2,L/2) (0,2R/π) (0,R/2) (0,h/4) (0,3R/8)

Question 5

com of extended body( from one extreme point )

Accepted Answer

m1r1+m2r2/m1+m2

Question 6

cavity in object

Accepted Answer

r(remaining)=-m(cavity)*r(cavity/M(remaining) r(remaining)=A1x1-A2x2/(A1-A2)(where x1 is original mass com for question x1 has to be set 0 x2 is removed cavity com)

Question 7

velocity of com vector form

Accepted Answer

V =(M1V1+M2V2+...)/M1+M2+M3+....

Question 8

acceleration of com vector form

Accepted Answer

a=(M1a1+M2a2+...)/M1+M2+M3...

Question 9

velocity of com in magnitude

Accepted Answer

v=√(((m1)^2(v1)^2+(m2)^2(v2)^2+2(m1v1m2v2)cos(theta))/m1+m2)

Question 10

isolated system

Accepted Answer

M1r1=M2r2

Question 11

moment of inertia

Accepted Answer

I=mr^2

Question 12

two point mass moment of inertia

Accepted Answer

I = m1r1^2+m2r2^2 r1=m2r/m1+m2 r2 =m1r/m1+m2 I= m(reduced)r^2 m(red)=m1m2/(m1+m2)

Question 13

moment of inertia about coordinate axes

Accepted Answer

r(x)=√(y^2+z^2) r(y)=√(x^2+z^2) r(z)=√(x^2+y^2) I(x)=mr(x)^2 etc

Question 14

moment of inertia of continuous mass

Accepted Answer

I =integral (r^2 dm)

Question 15

moment of inertia and axis passing through centre and perpendicular to planeon various objects (thin rod ,Ring ,disc , hollow cylinder,solid cylinder , hollow sphere, solid sphere)

Accepted Answer

Thin rod I=ml^2/12 ring=MR^2 disc=MR^2/2 , hollow cylinder=MR^2 ,solid cylinder=MR^2/2 hollow sphere=(2/3)MR^2 solid sphere=(2/5)MR^2

Question 16

theorem of perpendicular axis

Accepted Answer

Iz=Ix+Iy

Question 17

for 2D symmetric object

Accepted Answer

2I=Iz

Question 18

ring (parallel to plane) inertia disc(parallel to plane) square sheet (perpendicular to plane , parallel to plane) rectangular sheet (parallel to plane (sides l,b), perpendicular to plane)

Accepted Answer

Ring=MR^2/2 (parallel to plane)disc=MR^2/4(parallel to plane) square sheet (parallel to plane) =Ml^2/12 (perpendicular to plane )=Ml^2/6 rectangular sheet (parallel to plane )= Ml^2/12,Mb^2/12 (perpendicular to plane )=(M/12)(l^2+b^2)

Question 19

parallel axis theorem

Accepted Answer

I(parallel)=Icom+Ma^2

Question 20

moment of inertia for various objects (parallel axis) thin rod , ring , disc , hollow cylinder,solid cylinder, hollow sphere, solid sphere

Accepted Answer

thin rod =ml^2/3, ring (tangent perpendicular to plane) = 2MR^2 (tangent parallel to plane)= 3MR^2/2 disc (tangent perpendicular to plane)=3MR^2/2 (tangent parallel to plane)=5MR^2/4 hollow cylinder=2MR^2 solid cylinder=3MR^2/2 hollow sphere=5MR^2/3 solid sphere=7MR^2/5

Question 21

radius of gyration

Accepted Answer

I= Mk^2 (k is radius of gyration)

Question 22

annular Disc

Accepted Answer

I=(M/2)[R1^2+R2^2] here R1 is smaller radius and R2 longer radius

Question 23

torque

Accepted Answer

F×r =Frsin(theta)

Question 24

principle of moments

Accepted Answer

F1x1=F2x2

Question 25

angular acceleration

Accepted Answer

alpha=dw/dt=torque/I =mgsin(theta)l/I

Question 26

rod is released from Horizontal position (angular acceleration,linear acceleration of COM, Linear acceleration of point B at one end opposite from rigid end)

Accepted Answer

angular acceleration=3g/2L linear acceleration =3g/4 aB=3g/2

Question 27

translatory-rotation combination (angular acceleration, acceleration)

Accepted Answer

angular acceleration=mgR/(I(disc)+mR^2) acceleration=mg/m+(M/2) =2mg/(2m+M) M is mass of disc

Question 28

equation of rotational motion

Accepted Answer

w2=w1+alpha(t), theta=w1t+(1/2)alpha(t^2) , (w2)^2=(w1)^2+2aplha.theta, v=rw

Question 29

angular momentum (L)

Accepted Answer

L=r×p=m(r×v)=mvrsin(theta)

Question 30

spin angular momentum(if axis is at one end )

Accepted Answer

L=I(at that axis using parallel axis theorem)*w

Question 31

conservation of angular momentum

Accepted Answer

I1W1=I2W2

Question 32

if mass is constant what is the relation between T and R if density remains constant T and R

Accepted Answer

mass constant (T proportional to R^2 ) density constant (T proportional to R^5)

Question 33

when two bodies have w in same direction and w in opposite direction wfinial =?

Accepted Answer

wf(same direction)=I1W1+I2W2/(I1+I2) ,wf(different direction)=I1W1-I2W2/(I1+I2)

Question 34

Work =?

Accepted Answer

W=torque*theta

Question 35

Energy

Accepted Answer

K.E=(1/2)IW^2=L^2/2I=Lw/2

Question 36

work energy theorem

Accepted Answer

(1/2)I((W2)^2-(W1)^2)

Question 37

power(average, instantaneous)

Accepted Answer

P(average)=Work/time=∆K.E/time P(instantaneous)=torque*W (where W is omega)

Question 38

energy loss when 2 discs are joined (same direction w and different direction w)

Accepted Answer

same direction (∆K.E=I1I2/2(I1+I2)*(w1-w2)^2 different direction (∆K.E=I1I2/2(I1+I2)*(w1+w2)^2

Question 39

mechanical energy conservation (rod) U is mechanical energy w is omega also what is the equation for omega

Accepted Answer

Ui=MgL/2 Uf =(1/2)Iw^2 , w=√(3g/l)

Question 40

rolling motion (velocity at points vp)

Accepted Answer

Vp=v√(2(1+sin(theta))

Question 41

energy in rolling motion (translatory, rotational)

Accepted Answer

K.E(translatory)=(1/2)mv^2 K.E(rotational)=(1/2)Iw^2 =(1/2)mk^2*(v^2/R^2) K.E(total)=K.E(translatory)+K.E(rotational). K.Etotal/K.Etranslatory=(1+(k^2 /R^2) )

Question 42

motion along an inclined plane(velocity at bottom)

Accepted Answer

v=√(2gh/(1+(k^2/R^2))) v is proportional to 1/(k^2/R^2)

Question 43

motion along an inclined plane(acceleration)

Accepted Answer

gsin(theta)/(1+(k^2/R^2)) a is proportional to 1/(k^2/R^2)

Question 44

motion along an inclined plane(time of descend)

Accepted Answer

t=(1/sin(theta))*√((2h)*(1+(k^2/R^2)) t is proportional to (k^2/R^2)

Question 45

value of velocity (ring/hollow cylinder , disc/solid cylinder, hollow sphere, solid sphere)

Accepted Answer

ring/hollow cylinder = √gh disc/solid cylinder=√((4/3)gh) hollow sphere=√((6/5)gh). solid sphere= √((10/7)gh)

Question 46

Time to reach bottom

Accepted Answer

Ring>hollowsphere >Disc>solid sphere

Question 47

velocity comparison

Accepted Answer

solid sphere > Disc> hollow sphere> Ring

Physics: System of Particles and Center of Mass

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