A uniform solid right circular cone of base radius r is joined to a uniform solid hemisphere of radius r and of the same density, so as to have a common face. The centre of mass of the composite solid lies on the common face. The height of the cone is –
Answers
Answer:
Step 1:
Base radius of the right circular cone = r
Let the height of the cone be “h”.
∴ The volume of right circular cone = 1/3 * πr²h
And,
Mass of the cone, m₁ = density * volume = ρ * 1/3 * πr²h …… (i)
Step 2:
The radius of uniform solid hemisphere = r
The density of cone = density of hemisphere = ρ … [given]
∴ The volume of uniform solid hemisphere = ½ * 4/3 * πr³ = 2/3 * πr³
And,
Mass of the cone, m₂ = density * volume = ρ * 2/3 * πr³ …… (ii)
Step 3:
It is given that the centre of mass of the composite solid lies on the common face, therefore, we can say Ycm = 0.
The formula for the centre of mass of the combined system is given as,
Ycm = [m1y1 + m2y2] / [m1 + m2]
Substituting the values from eq. (i) & (ii), we get
⇒ 0 = [{ρ * 1/3 * πr²h * (h/4)}+{ ρ * 2/3 * πr³ * (-3r/8)}] / [{ρ * 1/3 * πr²h}+{ ρ * 2/3 * πr³}]
⇒ ρ * 1/3 * πr² [(h²/4) – (2r * 3r/8)] = 0
⇒ h²/4 – 3r²/4 = 0
⇒ h²/4 = 3r²/4
⇒ h = r*√3
Thus, the height of the cone is r√3.
Answer:
Step 1:
Base radius of the right circular cone = r
Let the height of the cone be “h”.
∴ The volume of right circular cone = 1/3 * πr²h
And,
Mass of the cone, m₁ = density * volume = ρ * 1/3 * πr²h …… (i)
Step 2:
The radius of uniform solid hemisphere = r
The density of cone = density of hemisphere = ρ … [given]
∴ The volume of uniform solid hemisphere = ½ * 4/3 * πr³ = 2/3 * πr³
And,
Mass of the cone, m₂ = density * volume = ρ * 2/3 * πr³ …… (ii)
Step 3:
It is given that the centre of mass of the composite solid lies on the common face, therefore, we can say Ycm = 0.
The formula for the centre of mass of the combined system is given as,
Ycm = [m1y1 + m2y2] / [m1 + m2]
Substituting the values from eq. (i) & (ii), we get
⇒ 0 = [{ρ * 1/3 * πr²h * (h/4)}+{ ρ * 2/3 * πr³ * (-3r/8)}] / [{ρ * 1/3 * πr²h}+{ ρ * 2/3 * πr³}]
⇒ ρ * 1/3 * πr² [(h²/4) – (2r * 3r/8)] = 0
⇒ h²/4 – 3r²/4 = 0
⇒ h²/4 = 3r²/4
⇒ h = r*√3
Thus, the height of the cone is r√3.