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Physics Numericals

Q11: (a) The diameter of neck and bottom of a bottle are 2 cm and 10 cm respectively. The bottle is completely filled with oil. If the cork in the neck is pressed in with a force of 1.2 kgf, what force is exerted on the bottom of the bottle?

Solution:
Given:
Diameter of neck: ( d_{text{neck}} = 2 , text{cm} ), Radius: ( r_{text{neck}} = frac{d_{text{neck}}}{2} = 1 , text{cm} )
Diameter of bottom: ( d_{text{bottom}} = 10 , text{cm} ), Radius: ( r_{text{bottom}} = frac{d_{text{bottom}}}{2} = 5 , text{cm} )
Force applied: ( F_{text{neck}} = 1.2 , text{kgf} = 1.2 times 10 , text{N} = 12 , text{N} )
Formula:
[
frac{F_{text{neck}}}{A_{text{neck}}} = frac{F_{text{bottom}}}{A_{text{bottom}}}
]
Area: ( A = pi r^2 )
Calculation:
Area of neck: ( A_{text{neck}} = pi (1)^2 = pi , text{cm}^2 )
Area of bottom: ( A_{text{bottom}} = pi (5)^2 = 25pi , text{cm}^2 )
[
F_{text{bottom}} = frac{F_{text{neck}} cdot A_{text{bottom}}}{A_{text{neck}}} = frac{12 cdot 25pi}{pi} = 300 , text{N}
]

(b) Name the law/principle you have used to find the force in part (a).

Answer: Pascal’s Law.

Q12: A force of 50 kgf is applied to the smaller piston of a hydraulic machine. Neglecting friction, find the force exerted on the large piston, if the diameters of the pistons are 5 cm and 25 cm respectively.

Solution:
Given:
Force on smaller piston: ( F_1 = 50 , text{kgf} = 50 times 10 = 500 , text{N} )
Diameter of smaller piston: ( d_1 = 5 , text{cm} ), Radius: ( r_1 = frac{d_1}{2} = 2.5 , text{cm} )
Diameter of larger piston: ( d_2 = 25 , text{cm} ), Radius: ( r_2 = frac{d_2}{2} = 12.5 , text{cm} )
Formula:
[
frac{F_1}{A_1} = frac{F_2}{A_2}
]
Area: ( A = pi r^2 )
Calculation:
Area of smaller piston: ( A_1 = pi (2.5)^2 = 6.25pi , text{cm}^2 )
Area of larger piston: ( A_2 = pi (12.5)^2 = 156.25pi , text{cm}^2 )
[
F_2 = frac{F_1 cdot A_2}{A_1} = frac{500 cdot 156.25pi}{6.25pi} = 12,500 , text{N}
]

Q13: Two cylindrical vessels fitted with pistons A and B of area of cross-section 8 cm² and 320 cm² respectively, are joined at their bottom by a tube and they are completely filled with water. When a mass of 4 kg is placed on piston A, find: (i) the pressure on piston A, (ii) the pressure on piston B, and (iii) the thrust on piston B.

Solution:
Given:
Mass on piston A: ( m = 4 , text{kg} )
Area of piston A: ( A_A = 8 , text{cm}^2 = 8 times 10^{-4} , text{m}^2 )
Area of piston B: ( A_B = 320 , text{cm}^2 = 320 times 10^{-4} , text{m}^2 )
Gravitational acceleration: ( g = 10 , text{m/s}^2 )
Formula:
Pressure: ( P = frac{F}{A} ), Thrust: ( F = P cdot A )
Calculations:
Force on piston A: ( F_A = m cdot g = 4 cdot 10 = 40 , text{N} )
(i) Pressure on piston A: ( P_A = frac{F_A}{A_A} = frac{40}{8 times 10^{-4}} = 0.5 , text{kgf/cm}^2 )
(ii) Pressure on piston B (same as ( P_A ) since they are connected): ( P_B = 0.5 , text{kgf/cm}^2 )
(iii) Thrust on piston B: ( F_B = P_B cdot A_B = 0.5 cdot 320 = 160 , text{N} )

Q14: What force is applied on a piston of area of cross-section 2 cm² to obtain a force of 150 N on the piston of area of cross-section 12 cm² in a hydraulic machine?

Solution:
Given:
Force on larger piston: ( F_2 = 150 , text{N} )
Area of larger piston: ( A_2 = 12 , text{cm}^2 )
Area of smaller piston: ( A_1 = 2 , text{cm}^2 )
Formula:
[
frac{F_1}{A_1} = frac{F_2}{A_2}
]
Calculation:
[
F_1 = frac{F_2 cdot A_1}{A_2} = frac{150 cdot 2}{12} = 25 , text{N}
]
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