A 1,500-kg car stands on a square platform, and a 25-kg boy stands on another square platform. These platforms are the two caps of a large oil-filled container. What must be the ratio of the platform sides (large/small) for the boy to gently lift the car?
To find the ratio of the platform sides (large/small) for the boy to gently lift the car, we need to consider the principles of buoyancy and equilibrium.
First, let's understand the concept of buoyancy. When an object is immersed in a fluid (in this case, oil), it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the object. This principle is known as Archimedes' principle.
In this situation, the small platform (on which the boy stands) must exert an upward force equal to the weight of the car to lift it. The large platform (beneath the car) exerts a downward force equal to the weight of the car.
To lift the car, the buoyant force acting on the car must equal its weight. The buoyant force on an object is given by the formula:
Buoyant force = Density of fluid x Volume of fluid displaced x Acceleration due to gravity
Since the density of oil and the acceleration due to gravity are constant, we can determine that the buoyant force depends on the volume of oil displaced by the car.
Given that the car has a mass of 1,500 kg, we can use the equation:
Weight of car = Mass of car x Acceleration due to gravity
Weight of car = 1,500 kg x 9.8 m/s^2
Now, let's analyze the boy's platform. For the boy to exert an upward force equal to the weight of the car, the weight of the oil displaced by the boy's platform must be equal to the weight of the car. We can represent this using the equation:
Weight of oil displaced by boy's platform = Weight of car
Now, let's consider each platform as a square plate with side lengths 'L' (large platform) and 'l' (small platform), respectively. The volume of oil displaced by each platform can be calculated by multiplying the area of the platform by the height.
The volume of oil displaced by the large platform is equal to:
Volume of oil displaced by large platform = L^2 x height
Likewise, the volume of oil displaced by the small platform is equal to:
Volume of oil displaced by small platform = l^2 x height
The height of the oil displaced by the platforms is irrelevant as long as it is constant, as it will cancel out when we take the ratio of the volumes.
To determine the ratio of the platform sides (large/small), we need to find the ratio of the volumes of oil displaced by each platform:
Ratio of platform sides (large/small) = (Volume of oil displaced by large platform) / (Volume of oil displaced by small platform)
Ratio of platform sides (large/small) = [(L^2 x height) / (l^2 x height)]
The height cancels out, leaving us with:
Ratio of platform sides (large/small) = L^2 / l^2
Now, substituting the weight equation for the car and the buoyant force equation, we get:
L^2 / l^2 = (Weight of car / (Density of oil x Volume of oil displaced by small platform))
Plugging in the values, we have:
L^2 / l^2 = (1500 kg x 9.8 m/s^2) / (Density of oil x l^2 x height)
Since we don't have the specific value of the density of oil or the height, we cannot determine the exact ratio of the platform sides (large/small) without this information. However, we have set up the equation in a way that allows you to substitute the known values to find the ratio once you have the additional information.