Question

You are preparing some apparatus for a visit to a newly discovered planet Caasi having oceans of glycerine and a surface acceleration due to gravity of 5.40 m/s\(^2\). If your apparatus floats in the oceans on earth with 25.0% of its volume submerged, what percentage will be submerged in the glycerine oceans of Caasi?

Short answer

19.8% of the apparatus's volume will be submerged in the glycerine oceans of Caasi.

Step-by-step solution

Understand Archimedes' Principle

To solve this problem, we must apply Archimedes' principle, which states that a floating object displaces a weight of fluid equal to its own weight. This principle is vital for calculating the percentage of the volume submerged in any fluid.

Calculate Submersion on Earth

On Earth, 25.0% of the apparatus's volume is submerged. This implies that the buoyant force equals the weight of the displaced fluid, which in this case is water. Let's denote the volume of the apparatus as \( V \). The weight of water displaced is equal to the weight of the apparatus. Therefore, \( 0.25 \times V \times \rho_{\text{water}} \times g = \text{Weight of apparatus} \).

Expression for Weight of Apparatus

Since on Earth the weight of the apparatus is balanced by the weight of the displaced water, write: \( 0.25 \times V \times \rho_{\text{water}} \times g = V_{\text{app}} \times g \), where \( V_{\text{app}} \) is the volume of the apparatus, \( \rho_{\text{water}} = 1000 \text{ kg/m}^3 \), and \( g = 9.81 \text{ m/s}^2 \). Thus, \( V_{\text{app}} = 0.25 \times V \times 1000 \).

Calculate Submersion on Caasi

On Caasi, the same apparatus floats in glycerine with surface gravity \( g' = 5.40 \text{ m/s}^2 \). Let \( x \times V \) be the submerged volume of apparatus. Then \( x \times V \times \rho_{\text{glycerine}} \times g' = V_{\text{app}} \times g' \). Using \( \rho_{\text{glycerine}} = 1260 \text{ kg/m}^3 \), we deduce \( x = \frac{V_{\text{app}}}{V \times 1260} \).

Final Calculation

Substitute \( V_{\text{app}} \) from earlier into \( x = \frac{0.25 \times V \times 1000}{V \times 1260} \) to find \( x = \frac{0.25 \times 1000}{1260} = \frac{250}{1260} \approx 0.198 \). Hence, about 19.8% of its volume will be submerged.

Key concepts

Buoyancy and Its Role in Floating

Buoyancy is the force that allows objects to float in fluids, such as liquids or gases. This force acts in the opposite direction to gravity, making things appear lighter in water or any fluid. This phenomenon is central to understanding Archimedes' Principle, which is essential when dealing with floating objects.

According to Archimedes' Principle, the buoyant force on an object submerged in fluid is equal to the weight of the fluid that the object displaces. If an object is partially submerged, it means that it displaces a volume of fluid whose weight equals the object's weight.

This principle helps us calculate how much of an object remains underwater based on its density and the fluid's density. If the buoyant force is greater than the object's weight, it will float. If lesser, it will sink. It’s fascinating to see how this principle applies to different fluids and gravitational conditions – like in the scenario with the planet Caasi.

Understanding Fluid Displacement

Fluid displacement happens when an object is submerged in a fluid, whether partially or fully. It pushes away or "displaces" a certain amount of that fluid. The displaced fluid's weight determines the upward buoyant force exerted on the object.

To determine how much of an object is submerged, you must consider the object's weight and the fluid's density. On Earth, the apparatus displaced enough water such that 25% of its volume was submerged, revealing the exact point where the force of gravity and the buoyancy force balance out.

• Denser objects displace more fluid to achieve buoyancy.
• If the fluid is denser, less volume needs to be displaced to equal the object's weight.
• This concept is crucial when calculating submersion percentages in different fluids or planets.

Experiments like these that involve different fluids and planetary conditions illustrate how versatile Archimedes' principle truly is.

Gravity Effects on Buoyancy

Gravity significantly impacts how buoyant an object is. While buoyant force mainly depends on fluid density and volume displaced, gravity dictates the weight of the displaced fluid, thereby affecting the floating balance.

The surface gravity on planets affects how objects float, as seen in the planet Caasi with a lower gravity than Earth (5.4 m/s\(^2\) compared to 9.81 m/s\(^2\)). This lower gravitational pull means that objects need to displace a relatively smaller volume of more dense fluid, like glycerine, to balance their weight.

• Stronger gravity would require more displaced volume to achieve buoyancy.
• The relationship between gravity and buoyancy forces determines submersion levels.
• Understanding these gravity effects helps in predicting how objects behave in different environments.

By accounting for gravity, you can better predict buoyancy and better prepare when working with floating devices on other planets.