One day when you come into physics lab you find several plastic hemispheres floating like boats in a tank of fresh water. Each lab group is challenged to determine the heaviest rock that can be placed in the bottom of a plastic boat without sinking it. You get one try. Sinking the boat gets you no points, and the maximum number of points goes to the group that can place the heaviest rock without sinking. You begin by measuring one of the hemispheres, finding that it has a mass of 21 g and a diameter of 8.0 cm. What is the mass of the heaviest rock that, in perfectly still water, won't sink the plastic boat?

Verified step by step guidance

1. Step 1: Calculate the volume of the plastic hemisphere. The formula for the volume of a hemisphere is \( V = \frac{2}{3} \pi r^3 \), where \( r \) is the radius. First, find the radius by dividing the diameter (8.0 cm) by 2.
2. Step 2: Determine the maximum buoyant force acting on the hemisphere. The buoyant force is equal to the weight of the water displaced by the hemisphere. Use the formula \( F_b = \rho_{water} \cdot V \cdot g \), where \( \rho_{water} \) is the density of water (approximately 1000 kg/m³), \( V \) is the volume of the hemisphere, and \( g \) is the acceleration due to gravity (9.8 m/s²).
3. Step 3: Calculate the total weight the hemisphere can support without sinking. This includes the weight of the hemisphere itself and the weight of the rock. The total weight is equal to the buoyant force calculated in Step 2.
4. Step 4: Subtract the weight of the hemisphere from the total weight calculated in Step 3 to find the maximum weight of the rock. Convert the mass of the hemisphere (21 g) into kilograms and use \( W = m \cdot g \) to find its weight.
5. Step 5: Convert the maximum weight of the rock into mass using \( m = \frac{W}{g} \), where \( W \) is the weight of the rock and \( g \) is the acceleration due to gravity. This will give the mass of the heaviest rock that can be placed in the boat without sinking it.

Key Concepts

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it. This force is equal to the weight of the fluid displaced by the object, as described by Archimedes' principle. For an object to float, the buoyant force must equal the weight of the object. In this scenario, the plastic hemisphere will float as long as the weight of the rock plus the weight of the hemisphere does not exceed the buoyant force acting on it.

Density

Density is defined as mass per unit volume and is a critical factor in determining whether an object will float or sink in a fluid. The density of the object must be less than the density of the fluid for it to float. In this case, the density of the plastic hemisphere and the water will influence how much mass can be added without exceeding the buoyant force, which is determined by the volume of water displaced.

Volume Displacement

Volume displacement refers to the amount of fluid that is displaced by an object when it is submerged. The volume of the hemisphere can be calculated using the formula for the volume of a sphere, adjusted for a hemisphere. This volume is crucial for calculating the buoyant force, as it determines how much water is displaced and, consequently, how much weight can be added to the hemisphere before it sinks.