The true weight of an object can be measured in a vacuum,...

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. An object of volume $V$ is weighed in air on a balance with the use of weights of density $\rho .$ If the density of air is $\rho_{\text {air }}$ and the balance reads $F_{g}^{\prime},$ show that the true weight $F_{g}$ is $$F_{R}=F_{R}^{\prime}+\left(V-\frac{F_{R}^{\prime}}{\rho g}\right) \rho_{\operatorname{ain}} g$$

The true weight of an object can be measured in a vacuum, where buoyant forces are absent. An object of volume $V$ is weighed in air on a balance with the use of weights of density $\rho .$ If the density of air is $\rho_{\text {air }}$ and the balance reads $F_{g}^{\prime},$ show that the true weight $F_{g}$ is
$$F_{R}=F_{R}^{\prime}+\left(V-\frac{F_{R}^{\prime}}{\rho g}\right) \rho_{\operatorname{ain}} g$$

Key Concepts

Archimedes' Principle

This principle states that any object fully or partially immersed in a fluid experiences an upward force, known as the buoyant force, equal to the weight of the fluid displaced by the object. It is the foundation for understanding how fluids affect the weight measurements of immersed bodies.

Buoyancy Correction in Weight Measurements

Due to the buoyant force exerted by air or any fluid, the weight measured in a non-vacuum environment (apparent weight) is less than the true gravitational weight. Correcting for buoyancy involves adding the effective weight of the displaced fluid back to the apparent weight, yielding the true weight of the object.

Density

Density, defined as mass per unit volume, is crucial for calculating the buoyant force. Both the density of the object (via its volume) and the density of the surrounding fluid are used to determine the weight of the displaced fluid, which is essential for applying buoyancy corrections in precise measurements.

True vs Apparent Weight

True weight is the gravitational force acting on an object in a vacuum where there are no buoyant effects, while apparent weight is the reading obtained from a balance in a fluid medium like air. The difference arises because the buoyant force counteracts part of the gravitational force, necessitating adjustments to obtain the object's true weight.

Transcript

00:01 Hi, so in this problem we have an object of volume v.

00:06 And we're trying to measure the true weight of this object by using a scale.

00:13 And we balance the scale with an object with density row.

00:18 We measure this weight as fg prime.

00:23 We're trying to figure out what the real true weight fg is.

00:27 Now you might be wondering, why isn't fg equal to fg prime? if the scale has balanced out.

00:34 Well, this is because there's some kind of a point force, b prime and also point force acting on object we're trying to measure, which is just going to, we're going to denote it just as b.

00:47 In order to figure out fg, we need to take into account these point forces.

00:54 So how do we do that? well, since the balance is scaled out, we can say that the difference between the gravitational force and the buoyant force for both sides are equal to each other.

01:10 So we get this expression.

01:12 Okay.

01:13 We're trying to find the true weight, so that's pretty simple.

01:16 We can bring b to the other side.

01:20 We can get this.

01:23 Now, we know fg prime, but we don't know what the buoyant forces are.

01:30 We need to figure out in terms of the variables we are given.

01:35 So let's try doing that.

01:37 Well, b is pretty simple.

01:40 It's just going to equal the volume of our object times the density that we're displacing, which is air times gravitational acceleration.

01:51 Well, what's b prime? well, same thing.

01:54 The volume of our weight times the air times gravitational acceleration...