A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water. A sphere, of radius 4 cm is attached to the top of the cone with a light inelastic string. The relative density of the cone is 1.8 and the relative density of the sphere is 0.7. Both the cone and the sphere are fully submerged in water and are at rest. (i) Show on separate diagrams the forces acting on the cone and on the sphere. (ii) Write equations to represent all the forces acting on the cone and the sphere. (iii) Calculate the tension in the string. (iv) Calculate the value of the reaction force between the base of the cone and the bottom of the tank. The pressure at the top of the sphere is 4000 Pa less than the pressure at the bottom of the tank. (v) Calculate the length of the string. (Density of water is 1000 kg m^-3)

Leaving Cert Applied Maths Hydrostatics: A right circular cone of radius

A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water - Leaving Cert Applied Maths - Question 9 - 2021

Question 9

A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water. A sphere, of radius 4 cm is attached to the top of the cone with a... show full transcript

Worked Solution & Example Answer:A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water - Leaving Cert Applied Maths - Question 9 - 2021

Step 1

Show on separate diagrams the forces acting on the cone and on the sphere.

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Answer

For the cone:

Weight (W1) acting downwards: W1 = ρ_cone × V_cone × g, where ρ_cone = 1.8 × 1000 kg/m³ and V_cone =

$V_{ ext{cone}} = rac{1}{3} imes ext{Area} imes ext{Height} = rac{1}{3} imes ( rac{22}{7} imes (0.12)^2) imes 0.25$

For the sphere:

Upthrust (B2) and weight (W2) acting downwards: W2 = ρ_sphere × V_sphere × g, where ρ_sphere = 0.7 × 1000 kg/m³.

Step 2

Write equations to represent all the forces acting on the cone and the sphere.

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Answer

For the cone:

Vertical forces: R + W1 = B1

For the sphere:

Vertical forces: T + B2 = W2

Step 3

Calculate the tension in the string.

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Answer

From the equations, substituting for W1 and W2, we have:

B2 = T + W2
Considering relative density and volumes, 1000g = T + 700g
Solve for T: $T = 700g - 1000g = 0.80 ext{ N}$

Step 4

Calculate the value of the reaction force between the base of the cone and the bottom of the tank.

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Answer

Using the pressures:

P1 = R + 0.80 + 1000g
Setting P2 as the pressure at the bottom of the tank, we find: $R + 0.80 + 1000 imes 9.81 = 1800g$
Solve for R: $R = 29.36 ext{ N}$

Step 5

Calculate the length of the string.

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Answer

Using the pressure differential:

P1 + P2 = 4000 ext{ Pa}
$1000 imes 0.25 + 1000 imes 10 imes h_2 = 4000 ext{ Pa}$ Resulting in:
Finding h2 leads to the total length L = 0.07 m.

A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water - Leaving Cert Applied Maths - Question 9 - 2021

Worked Solution & Example Answer:A right circular cone of radius 12 cm and height 25 cm is placed at the bottom of a tank of water - Leaving Cert Applied Maths - Question 9 - 2021

Show on separate diagrams the forces acting on the cone and on the sphere.

For the sphere:

Write equations to represent all the forces acting on the cone and the sphere.

Calculate the tension in the string.

Calculate the value of the reaction force between the base of the cone and the bottom of the tank.

Calculate the length of the string.

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