Solve this:
A carpenter has constructed a toy as shown in the adjoining figure. If the density of the material of the sphere is 12 times that of cone, the position of the centre of mass of the toy is given by :
(1) at a distance of 2R from O
(2) at a distance of 3R from O
(3) at a distance of 4R from O
(4) at a distance of 5R from O
Answer should be 3.
Centre of mass of a solid cone is at a distance of H/4 from its base =4R/4=R .(from O)
and the centre of mass of a sphere is at its centre = 4R+R=5R
If we take the density of cone to be x , then density of sphere is 12x .
Mass of cone = Volume * Density = 1/3pir^h*x = 1/3*pi*((2R)^2)*4R*x = 16/3pi(R^3)x
Mass of sphere = Volume * Density =4/3pi(R^3)*12x= 16pi(R^3)x
Now using the formula for centre of mass , y = (M(Sphere)*COM(sphere) + M(Cone)*COM(cone))/(M(cone) + M(sphere))
where COM= y co-ordinate of centre of mass and M= mass.
y= 16pi(R^3)x*5R + 16/3pi(R^3)x*R/16(piR^3)x +16/3pi(R^3)
= (80+(16/3))R/ (16 +(16/3))
=256R/64
=4R
Centre of mass of a solid cone is at a distance of H/4 from its base =4R/4=R .(from O)
and the centre of mass of a sphere is at its centre = 4R+R=5R
If we take the density of cone to be x , then density of sphere is 12x .
Mass of cone = Volume * Density = 1/3pir^h*x = 1/3*pi*((2R)^2)*4R*x = 16/3pi(R^3)x
Mass of sphere = Volume * Density =4/3pi(R^3)*12x= 16pi(R^3)x
Now using the formula for centre of mass , y = (M(Sphere)*COM(sphere) + M(Cone)*COM(cone))/(M(cone) + M(sphere))
where COM= y co-ordinate of centre of mass and M= mass.
y= 16pi(R^3)x*5R + 16/3pi(R^3)x*R/16(piR^3)x +16/3pi(R^3)
= (80+(16/3))R/ (16 +(16/3))
=256R/64
=4R