Physics: System of Particles and Center of Mass

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

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

r =integral (dm *r)/integral dm

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

m1r1+m2r2/m1+m2

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)

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

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

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

M1r1=M2r2

I=mr^2

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

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

I =integral (r^2 dm)

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

Iz=Ix+Iy

2I=Iz

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)

I(parallel)=Icom+Ma^2

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

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

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

F×r =Frsin(theta)

F1x1=F2x2

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

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

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

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

L=r×p=m(r×v)=mvrsin(theta)

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

I1W1=I2W2

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

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

W=torque*theta

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

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

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

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

Ui=MgL/2 Uf =(1/2)Iw^2 , w=√(3g/l)

Vp=v√(2(1+sin(theta))

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) )

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

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

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

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

Ring>hollowsphere >Disc>solid sphere

solid sphere > Disc> hollow sphere> Ring