A tall glass (20cm) is floating in your bathtub. It weighs 60 grams and has a bottom area of 10cm^2. How deep will it sink into the water, while it is still empty and how much water can you add to make it sink?
To determine how deep the empty glass will sink into the water, we need to consider the principle of buoyancy.
According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, since the glass is empty, it displaces an amount of water equal to its own volume when submerged.
To calculate the depth to which the empty glass will sink, we need to determine its volume. The volume of a shape with a circular base, like the glass, can be calculated using the formula V = πr^2h, where r is the radius of the circular base and h is the height of the shape.
Given that the glass has a bottom area of 10cm^2, we can use this to find the radius. The formula for the area of a circle is A = πr^2. Rearranging the formula, we have r = √(A/π), where A is the area of the circle. Plugging in the given value, we find r = √(10/π) cm.
Since the glass is empty, its volume is equal to the volume of water it will displace. When the glass is fully submerged, its volume will be equal to the volume of the cylinder formed by the water depth (h) and the circular base. The volume formula can be rearranged as h = V/(πr^2). Plugging in the values we have, h = V/(π * (√(10/π))^2) cm.
Now, to determine how much water needs to be added for the glass to sink, we need to find the maximum weight the glass can displace before it sinks. This weight is equal to the weight of the water that fills the glass completely.
Considering that the glass weighs 60 grams and assuming it is made of a material with the same density as water (1 g/cm^3), the volume of the glass will be equal to its weight (60 grams) in cm^3.
To find how much water needs to be added for the glass to sink, we subtract the volume of the glass from the volume of the glass when it is fully submerged (calculated earlier).
So, the answer to the first part of the question, about the depth to which the empty glass will sink, is h = V/(π * (√(10/π))^2) cm, where V is the volume of the glass.
The answer to the second part, about the amount of water to be added for the glass to sink, is the volume of the glass when it is fully submerged (calculated earlier) minus the volume of the glass itself (60 grams).