A body floats in a liquid contained in a beaker. The whole system as shown in figure falls freely un

Q: If upthrust is zero, why doesn't the body sink to the bottom of the beaker?

In free fall, the effective gravity acting on the body is also zero. Since there is no downward gravitational force pulling the body down, it has no tendency to sink. Both gravity and upthrust vanish, leaving the body suspended in its original position.

Q: Does this mean Archimedes' principle fails in free fall?

No, Archimedes' principle still holds perfectly. The principle states that upthrust equals the weight of the displaced liquid. In free fall, the displaced liquid is weightless, so its weight is zero, which correctly yields an upthrust of zero.

Q: What if the beaker was accelerated upwards instead of falling freely?

If the beaker accelerates upwards with an acceleration $a$, the effective gravity increases to $g_{\text{eff}} = g + a$. Consequently, the pressure gradient increases, and the upthrust on the body also increases.

Q: Is the pressure at the bottom of the beaker zero during free fall?

No, the absolute pressure is not zero. The gauge pressure (pressure due to the liquid column) becomes zero everywhere, meaning the pressure at every point in the liquid is simply equal to the atmospheric pressure $P_0$ acting on the surface.

Q: Does this phenomenon apply to gases as well, like a hot air balloon in free fall?

Yes, absolutely! Any fluid (whether liquid or gas) relies on gravity to create a pressure gradient. In free fall, the pressure gradient disappears in both liquids and gases, meaning all buoyant forces (upthrust) vanish completely.

Question

A body floats in a liquid contained in a beaker. The whole system as shown in figure falls freely under gravity. The upthrust on the body is

Options:

(A) zero

(B) equal to the weight of liquid displaced

(C) equal to the weight of the body in air

(D) equal to the weight of the immersed portion of the body

Sigma Percentile · Accepted Answer

### The Magic of Floating

Imagine you are standing next to a calm lake, watching a wooden block float peacefully on the water.

It seems so simple, yet beneath the surface, a beautiful tug-of-war of physics is taking place.

On one hand, gravity is relentlessly pulling the block downward with a force equal to its weight, $W = Mg$.

On the other hand, the water is pushing back upward with a force we call **buoyancy** or **upthrust**, $F_B$.

Under normal conditions, these two forces are in perfect harmony, keeping the block in a state of stable equilibrium.

But what happens if we suddenly cut the thread of gravity?

What if we take this entire setup—the beaker, the water, and the floating block—and drop it from a high tower, letting it fall freely?

Does the block sink? Does it shoot out of the water? Or does something even more fascinating happen?

Let's dive deep into the physics of free fall and fluid dynamics to find out.

### The Physics of Upthrust

To understand what happens in free fall, we must first understand where upthrust actually comes from.

It is a common misconception that fluids simply "dislike" submerged objects and push them up.

In reality, upthrust is the result of a **pressure difference**.

Because of gravity, the pressure in a static fluid increases with depth.

This variation is mathematically described by the hydrostatic pressure gradient equation:

$$\frac{dP}{dh} = ho_L g$$

Here, $ho_L$ is the density of the liquid, $h$ is the depth, and $g$ is the acceleration due to gravity.

Because of this gradient, the pressure at the bottom of a submerged object is always greater than the pressure at its top.

This pressure difference creates a net upward force, which we call the buoyant force:

$$F_B = V_i ho_L g$$

where $V_i$ is the volume of the submerged portion of the body.

Notice how both equations are directly proportional to $g$.

This is our crucial clue!

### Plunging into Free Fall

Now, let's release the entire system into a state of **free fall**.

As the beaker, liquid, and body plunge downwards, they accelerate at exactly the rate of gravity, $a = g$.

If we step inside the accelerating frame of reference of the beaker, we must use the concept of **effective gravity** ($g_{	ext{eff}}$):

$$g_{	ext{eff}} = g - a$$

Since the system is falling freely, the downward acceleration of our frame is exactly equal to $g$.

Substituting $a = g$ into our equation, we get:

$$g_{	ext{eff}} = g - g = 0$$

In this freely falling frame, the effective gravity is exactly **zero**!

The entire system enters a state of complete weightlessness.

### The Disappearing Pressure Gradient

Now, let's see what happens to our pressure gradient when $g_{	ext{eff}} = 0$.

Substituting this value back into our hydrostatic equation:

$$\frac{dP}{dh} = ho_L g_{	ext{eff}} = ho_L (0) = 0$$

This is a profound realization!

Since the pressure gradient is zero, the pressure does not change with depth anymore.

The pressure is completely uniform and equal to the atmospheric pressure $P_0$ at every single point in the liquid.

Without a pressure difference between the top and bottom of our submerged body, there is no longer any net upward force from the liquid.

We can also see this directly from Archimedes' formula by substituting $g_{	ext{eff}} = 0$:

$$F_{B,	ext{eff}} = V_i ho_L g_{	ext{eff}} = 0$$

Thus, the upthrust acting on the body during free fall is exactly **zero**.

This perfectly matches **Option (a)**.

### What Happens to the Floating Body?

Let's take a moment to appreciate the physical reality of this state.

Since the upthrust is zero, you might initially think the body should sink.

But remember, in this weightless environment, the effective gravity pulling the body down is also zero!

Both the downward gravitational pull and the upward buoyant force have completely vanished.

The net force acting on the body is exactly zero.

As a result, the body will simply remain in its original position relative to the liquid, suspended in a state of perfect weightlessness.

It won't sink, and it won't rise; it will just float effortlessly, as if suspended in deep space.

Animated Solution for Physics - Properties of Solids and Liquids: A body floats in a liquid contained in a beaker. The whole system as shown in figure falls freely under gravity. The upthrust on the body is

Select Answer:

Visualized Solution

Understanding the Floating Body Setup

The Origin of Upthrust

Hydrostatic Pressure Variation

Introducing Free Fall

Effective Gravity in Free Fall

Pressure Gradient Becomes Zero

Calculating the Upthrust

Conclusion

What Happens to the Body?

The Sigma Insight: Buoyancy and Archimedes' Principle

The Magic of Floating

The Physics of Upthrust

Plunging into Free Fall

The Disappearing Pressure Gradient

What Happens to the Floating Body?

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