Question

Advertisements for a certain small car claim that it floats in water. (a) If the car's mass is 900 kg and its interior volume is 3.0 m\(^3\), what fraction of the car is immersed when it floats? Ignore the volume of steel and other materials. (b) Water gradually leaks in and displaces the air in the car. What fraction of the interior volume is filled with water when the car sinks?

Short answer

(a) 30% of the car is immersed. (b) 30% of the interior volume is filled with water when it sinks.

Step-by-step solution

Understanding Archimedes' Principle

Archimedes' principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. For something floating, this buoyant force is equal to the weight of the object itself.

Calculating the Buoyant Force

The weight of the car is calculated using the formula \( F_g = m \cdot g \), where \( m = 900 \) kg and \( g = 9.8 \) m/s extsuperscript{2}. Thus, \( F_g = 900 \times 9.8 = 8820 \) N. This is also the buoyant force when the car is floating.

Calculating Water Displacement

The buoyant force (8820 N) is equal to the weight of the water displaced. Using \( F_b = \rho_{water} \cdot V_{displaced} \cdot g \), where \( \rho_{water} = 1000 \) kg/m extsuperscript{3}, we get \( V_{displaced} = \frac{F_b}{\rho_{water} \cdot g} = \frac{8820}{1000 \times 9.8} = 0.9 \) m extsuperscript{3}.

Fraction of Car Immersed

The immersed fraction of the car is \( \frac{V_{displaced}}{V_{car}} = \frac{0.9}{3.0} = 0.3 \). Therefore, 30% of the car's volume is immersed when it floats.

Understanding Sinking Condition

When the car sinks, the weight of water displaced equals the total volume of the car multiplied by the density of water and gravitational acceleration. To fully displace the entire volume of the car, the water inside the car must equal the car's mass.

Calculating Volume for Sinking

As the car sinks when it holds an amount of water equivalent in weight to the total weight of the car, this can be expressed as \( V_{water} \times \rho_{water} \times g = m \cdot g \). Hence, \( V_{water} = \frac{m}{\rho_{water}} = \frac{900}{1000} = 0.9 \) m\(^3\).

Fraction of Interior Filled with Water at Sinking

The fraction of the interior volume filled with water is \( \frac{V_{water}}{V_{car}} = \frac{0.9}{3.0} = 0.3 \). Thus, 30% of the interior volume is filled with water when the car sinks.

Key concepts

Buoyant Force

The concept of buoyant force is key to understanding why objects float or sink in a fluid. According to Archimedes' Principle, the buoyant force acting on an object is directly equal to the weight of the fluid that the object displaces while it is immersed. This principle is what allows a small car or even a massive ship to float on water.
For an object to float, the buoyant force must balance the weight of the object. Take the car in the provided exercise, the weight of the car is 8820 N, which is exactly the buoyant force when it is floating. This equilibrium is crucial; if the weight of the object is greater than the buoyant force, the object will sink, whereas if it is less, the object will float higher in the water.
Thus, the key takeaway is that the buoyant force is dependent on the volume of fluid displaced and is necessary to counteract the gravitational pull on the object.

Water Displacement

Water displacement is the volume of water pushed aside by an object submerged in it. This concept goes hand in hand with buoyant force. When an object is placed in water, it displaces an amount of water equal to the volume of the object's submerged part.
In the case of the car described in the exercise, 0.9 cubic meters of water is displaced when it is floating. This displaced volume corresponds to the weight of the car, which is held by the buoyant force. As a result, the volume of water displaced determines the upward force on the object.
Understanding water displacement is essential when calculating how much of an object stays above water, and how much is submerged. It becomes evident that as more of the object displaces water, the buoyant force increases until it can no longer keep the object afloat, causing it to sink.

Density of Water

Density plays a significant role in Archimedes' Principle and impacts buoyancy significantly. The density of water is typically around 1000 kg/m³, and this value is critical in determining whether an object will float or sink.
Objects with a density greater than water tend to sink, while those with less density float. This is because holding the same volume of water would weigh less than the object with higher density. For the car in the exercise, despite its mass, its larger interior volume filled with air allows it to initially float.
When considering the volume of water a car can displace as seen in the exercise, this density value is used to determine which fraction of the object can be submerged before sinking occurs. The calculations show that both the displaced volume and needed volume to fill equate to the car's weight, demonstrating how crucial the understanding of density and displacement is in buoyancy problems.