A cubical metal box has side L=10.34cm and mass m=150 g. How much mass must be placed inside the box if it is to float in water with its top at the surface?
volume of water displaced = .1034^3 = .0011055 m^3
times density of water which is about 1000 kg/m^3
=1.106 kg of water displaced
Archimedes says we can float that much mass
.150 + m = 1.106
m = .956 kg = 956 grams
To determine how much mass must be placed inside the box for it to float in water with its top at the surface, we need to consider the concepts of buoyancy and density.
Buoyancy is the upward force exerted on an object immersed in a fluid (in this case, water), which opposes the force of gravity. For an object to float, the buoyant force must be equal to or greater than the weight of the object.
The weight of the object can be calculated using the formula:
Weight = mass * gravitational acceleration (g)
Given that the mass of the box is 150 g, we first need to convert it to kilograms since the SI unit of mass is kilograms.
Mass = 150 g = 150/1000 kg = 0.15 kg
The gravitational acceleration (g) is approximately 9.8 m/s^2.
Weight = 0.15 kg * 9.8 m/s^2 = 1.47 N
Now, let's consider the density of water. The density of water at room temperature is approximately 1000 kg/m^3.
The buoyant force can be calculated using the formula:
Buoyant force = density of fluid * volume of displaced fluid * gravitational acceleration (g)
Since the box is floating at the surface, it displaces its own volume of water. The volume of the box can be calculated using the formula for the volume of a cube:
Volume of cube = side length^3 = L^3
Given that the side length (L) is 10.34 cm, we need to convert it to meters.
L = 10.34 cm = 10.34/100 m = 0.1034 m
Volume of cube = 0.1034 m * 0.1034 m * 0.1034 m = 0.00109968 m^3
Buoyant force = 1000 kg/m^3 * 0.00109968 m^3 * 9.8 m/s^2 = 10.79 N
To float with its top at the surface, the buoyant force must equal or exceed the weight of the box. Therefore, to find the additional mass needed, we can subtract the weight of the box from the buoyant force:
Additional mass = (Buoyant force - Weight) / g
Additional mass = (10.79 N - 1.47 N) / 9.8 m/s^2 = 1.07 kg
Therefore, an additional mass of 1.07 kg must be placed inside the box for it to float in water with its top at the surface.