A hollow plastic sphere is held below the surface of a...

A hollow plastic sphere is held below the surface of a fresh- water lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 $\mathrm{m}^{3}$ and the tension in the cord is 900 $\mathrm{N} .$ (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere? (c) The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

Key Concepts

Archimedes' Principle

This principle states that any object immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. It is fundamental in understanding why objects float or sink and is used to calculate the buoyant force acting on bodies in fluids.

Buoyant Force

Buoyant force is the upward force exerted by a fluid on a submerged or partially submerged object. It is calculated as the product of the fluid's density, gravitational acceleration, and the volume of fluid displaced by the object, which is key to analyzing equilibrium in fluid mechanics.

Force Equilibrium in Fluids

In fluid statics, analyzing force equilibrium involves balancing the downward gravitational force on an object with the upward buoyant force and any additional forces such as tension. This equilibrium condition is critical for determining whether an object sinks, floats, or remains in static equilibrium.

Density and Specific Weight

Density, defined as mass per unit volume, and specific weight, defined as weight per unit volume, are crucial in calculating buoyancy. The density of the fluid relative to the object’s density determines the magnitude of the buoyant force and the buoyancy behavior of the object.

Submerged Fraction in Floating Equilibrium

When an object floats, the fraction of its volume that remains submerged is determined by the balance between its weight and the buoyant force it experiences. This concept allows for the prediction of how much of an object's volume will be below the surface when in floating equilibrium, based on the ratio of the object's average density to the fluid's density.

Transcript

00:01 Okay, so in this problem we have a sphere attached to a string or a rope that is fixed on the bottom of a lake.

00:15 So this sphere is underwater.

00:29 Okay, so the first thing we need to calculate here is what is the buoyant force.

00:35 So what are the forces involved in this problem? first of all, we have the tension on the rope, that's called t.

00:42 We have actually the weight of the sphere which is m g and we have the buoyant force which maintains the system in equilibrium so to calculate the buoyant force we just need to use the system so the buoyant force fb is going to be equal the tension oops the tension plus the weight just going to be the tension plus m g so what do we actually know? well, we don't actually need to calculate this because we don't have the tension.

01:26 And we don't, so if we don't have the tension, we can calculate the buoyant force.

01:32 But if we remember the akhmat's principle, we know that the buoyant force can be described, as put in here, the buoyant force can be described as the density of the liquid, which in this case is the water times the volume of the object that is submerged, summerge times the gravity acceleration.

02:01 So that's our equation that we can calculate.

02:05 So let's do this.

02:06 The buoyant force is going to be equal the density of the water, 1000 times the volume of the object which is 0 .65 times the gravity is acceleration which is 9 .8 actually let's use just 8 not 81 9 .8 so calculating this we have a buoyant force of 6 .37 times 10 to the 3 newtons so that's the first answer now let's calculate the mass of these sphere so now we can use this equation here so we know that the buoyant force is equal the tension plus the weight so we can say that the mass of the sphere is going to be equal the buoyant force minus the tension divided by the gravity acceleration so this is going to be equal let's see six point 37 times 10 to the 3 minus 900 divided by 9 .8.

03:46 So if we calculate this we have the mass of the sphere equals 5, 5, 8 kilograms.

03:58 Okay so and finally the last thing we need to calculate is when this sphere comes to rest, what fraction of his volume will be submerged.

04:10 So let's see.

04:12 Let's put the drawing here.

04:14 In the final moment, we have here the water, and we have here the sphere...

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