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A metallic sphere weighting 3 kg in air is held by a string so as to be completely immersed in a liquid of relative density 0.8. The relative density of metallic is 10. The tension in the string is
The tension in the string is 27.6 N.
To find the tension in the string, we need to analyze the forces acting on the metallic sphere when it’s completely immersed in the liquid.
Weight of the sphere in air:
W_sphere = m_sphere * g
Here, m_sphere = 3 kg, and g = 10 m/s^2.
So, W_sphere = 3 * 10 = 30 N.
Buoyant force on the sphere:
F_buoyant = V_sphere * ρ_liquid * g
Here, V_sphere is the volume of the sphere,
ρ_liquid is the density of the liquid, and g = 10 m/s^2.
The volume of the sphere can be calculated using its mass and the density of the metal:
V_sphere = m_sphere / ρ_metal
V_sphere = 3 kg / 10 = 0.3 m^3
Given that ρ_liquid = 0.8,
we calculate:
F_buoyant = 0.3 * 0.8 * 10 = 2.4 N
Now, since the sphere is completely immersed and there is no acceleration in the vertical direction, the tension in the string must balance the net vertical force acting on the sphere. Therefore,
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T = W_sphere – F_buoyant
T = 30 N – 2.4 N = 27.6 N
So, the tension in the string is 27.6 N.
Define Buoyant Force
The buoyant force, also known as buoyancy, is a fundamental principle in fluid dynamics that describes the upward force exerted on an object immersed in a fluid (liquid or gas). This force opposes the weight of the object and is a result of the pressure difference between the top and bottom of the object when submerged in the fluid. The concept was first discovered by the ancient Greek mathematician and physicist Archimedes, hence often referred to as “Archimedes’ principle.”
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When an object is immersed in a fluid, it displaces a volume of the fluid equal to its own volume. The buoyant force acting on the object is equal to the weight of the displaced fluid. Mathematically, this can be expressed by Archimedes’ principle equation:
Buoyant force = Weight of displaced fluid
This principle is crucial in understanding various phenomena, such as why objects float or sink in fluids and how ships and submarines operate.
The buoyant force arises due to the difference in pressure between the top and bottom of an immersed object. The pressure at the bottom of the object is higher than at the top because of the weight of the fluid above it. This pressure difference results in a net upward force exerted on the object, which is the buoyant force.
The magnitude of the buoyant force depends on the density of the fluid, the volume of fluid displaced, and the acceleration due to gravity. Specifically, the formula for calculating the buoyant force is:
Buoyant force = Density of fluid × Volume of displaced fluid × Acceleration due to gravity
In simpler terms, the buoyant force increases with the density of the fluid and the volume of the object immersed in it. Objects with greater volume relative to their weight tend to experience a larger buoyant force, making them more likely to float.
Buoyancy plays a crucial role in various applications, including shipbuilding, aviation, and engineering. Engineers and designers must consider buoyancy when designing objects to ensure they have the necessary stability and buoyant force to operate effectively in fluids.
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Overall, the buoyant force is a fundamental concept in fluid mechanics that explains the behavior of objects submerged in fluids and is essential for understanding numerous natural and engineered systems.
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