rho=2r^2 Find centre of mass & Solid Hemisphere
Answers
Answer:
zop tq cp cs yar kya kr rhe hai to the new phone number
Answer:
We are considering a solid hemisphere of mass M and has the radius R. The centre of mass will lie on the vertical line passing through the centre of the hemisphere, the vertical line is also the normal to the base. In order to find the centre of mass, we have to consider an element.
We are taking an elemental disc at a height h from the base of the hemisphere. The mass of the elemental disc is dM and the width is dy.
The radius of the disc is r=\sqrt{R^{2}-y^{2}}r=
R
2
−y
2
—-(1)
Mass of the disc dM = (3M/2πR3) x (πr2dy)———————-(2)
Substitute equa(1) in equa(2)
dM = (3M/2πR3) x π(R2-y2)dy)
Y-coordinate of Centre of mass,yc= (1/M)ഽydM,
Here y is the y-coordinate which represents the height of the elemental disc from the base.
Putting the value of dM and calculating the centre of mass, we get
yc= (1/M)ഽy(3M/2R3) x (R2-y2)dy)
Integrating between the limits 0⟶R
y_{c}=\frac{1}{M}\int_{0}^{R}\frac{3M}{R^{3}}(R^{2}-y^{2})dy\times yy
c
=
M
1
∫
0
R
R
3
3M
(R
2
−y
2
)dy×y y_{c}=\frac{3}{2R^{3}}\int_{0}^{R}{}(R^{2}y-y^{3})dyy
c
=
2R
3
3
∫
0
R
(R
2
y−y
3
)dy
= (3/2R2)[(R4/2) – (R4/4)] = 3R/8
yc = 3R/8
Centre of Mass of the solid hemisphere, yc = 3R/8,
Here R is the radius of the hemisphere.
HOPE IT HELPS UH.