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2 (a) A cytinder is nuspended from the end of a string. The cyinder is stationa...

Sep 16, 2024

Solution

To calculate the height of the cylinder, we can use the difference in pressure between the top and bottom faces of the cylinder. The pressure difference is given as 320 Pa. The formula for pressure difference is given by:

\Delta P = \rho g h

where

\Delta P

is the pressure difference,

\rho

is the density of the fluid (water),

g

is the acceleration due to gravity (approximately

9.81 \, \text{m/s}^2

), and

h

is the height of the cylinder. Rearranging the formula gives us:

h = \frac{\Delta P}{\rho g} = \frac{320 \, \text{Pa}}{(1.0 \times 10^3 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)} \approx 0.0325 \, \text{m}

To calculate the upthrust (buoyant force) acting on the cylinder, we can use Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. The volume of the cylinder can be calculated using its height and cross-sectional area. The area

A

of the circular cross-section is given by:

A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{0.031 \, \text{m}}{2}\right)^2 \approx 7.54 \times 10^{-5} \, \text{m}^2

. The volume

V

of the cylinder is then:

V = A \cdot h \approx 7.54 \times 10^{-5} \, \text{m}^2 \cdot 0.0325 \, \text{m} \approx 2.45 \times 10^{-6} \, \text{m}^3

. The buoyant force

F_b

is then calculated as:

F_b = \rho V g = (1.0 \times 10^3 \, \text{kg/m}^3)(2.45 \times 10^{-6} \, \text{m}^3)(9.81 \, \text{m/s}^2) \approx 0.024 \, \text{N}

. However, the problem states that the upthrust is 0.38 N, which indicates that the cylinder is partially submerged

To find the tension

T

in the string, we can apply the equilibrium condition. The forces acting on the cylinder are its weight

W = 0.84 \, \text{N}

acting downwards, the buoyant force

F_b

acting upwards, and the tension

T

in the string also acting upwards. The equilibrium condition gives us:

T + F_b = W

. Rearranging this gives us:

T = W - F_b = 0.84 \, \text{N} - 0.38 \, \text{N} = 0.46 \, \text{N}

Answer

0.46 N

Key Concept

Archimedes' Principle: The buoyant force on an object is equal to the weight of the fluid displaced by the object. Equation:

F_b = \rho V g

(Buoyant force equals density times volume times acceleration due to gravity). Static Equilibrium: The net force acting on an object at rest is zero. Equation:

\sum F = 0

Explanation

The calculations show that the height of the cylinder is approximately 0.0325 m, the upthrust acting on the cylinder is 0.38 N, and the tension in the string is 0.46 N, confirming the equilibrium of forces acting on the cylinder.