A ball has a diameter of 3.89 cm and average density of 0.0842 g/cm3. What force is required to hold it completely submerged under water?
find the volume of the ball (sphere)
mass = volume * density
forces on the ball
... gravity ... downward ... m * g
... buoyancy ... upward ... equal to mass of water displaced times g
(ball volume) * (water density) * g =
... [(ball mass) * g] + (hold down force)
To calculate the force required to hold the ball completely submerged under water, we need to consider the buoyant force acting on the ball and the gravitational force acting on it.
Let's first find the volume of the ball:
The diameter of the ball is given as 3.89 cm, so the radius (r) can be calculated by dividing the diameter by 2:
r = 3.89 cm / 2 = 1.945 cm
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Substituting the radius into the formula:
V = (4/3) * π * (1.945 cm)^3
Calculating the volume:
V ≈ 21.66 cm^3
Now that we have the volume, we can calculate the mass of the ball:
The density of the ball is given as 0.0842 g/cm^3, and the volume is 21.66 cm^3. So, the mass (m) of the ball is given by:
m = density * volume
Substituting the values:
m = 0.0842 g/cm^3 * 21.66 cm^3
Calculating the mass:
m ≈ 1.825 g
Now, let's calculate the gravitational force acting on the ball:
The gravitational force (Fg) is given by:
Fg = m * g
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
Converting the mass to kilograms:
m ≈ 1.825 g / 1000 = 0.001825 kg
Calculating the gravitational force:
Fg = 0.001825 kg * 9.8 m/s^2
Fg ≈ 0.01787 N
Now, let's calculate the buoyant force acting on the ball:
The buoyant force (Fb) is given by:
Fb = ρf * V * g
where ρf is the density of the fluid (water), V is the volume of the ball, and g is the acceleration due to gravity.
The density of water is approximately 1000 kg/m^3, but we need to convert it to grams per cubic centimeter:
ρf = 1000 kg/m^3 * (1 g / 1000 kg) / (1 cm/10 mm)^3 ≈ 1 g/cm^3
Substituting the values:
Fb = (1 g/cm^3) * 21.66 cm^3 * 9.8 m/s^2
Calculating the buoyant force:
Fb ≈ 211.848 N
Since the ball is completely submerged in water and at rest, the buoyant force acting upwards must be equal to the gravitational force acting downwards. Therefore, the force required to hold the ball completely submerged under water is equal to the gravitational force:
The force required = 0.01787 N
To find the force required to hold a ball completely submerged under water, we can use the principles of buoyancy. Buoyancy is the upward force exerted by a fluid on a submerged object.
The force of buoyancy can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the submerged object. This can be expressed mathematically as:
Buoyant force = Weight of displaced fluid
The weight of the fluid can be found using the density of the fluid and the volume of the fluid displaced by the submerged object. In this case, the fluid is water.
To find the volume of the fluid displaced, we need to determine the volume of the ball. The volume of a sphere can be calculated using the formula:
Volume of a sphere = (4/3) * π * radius^3
Given that the diameter of the ball is 3.89 cm, the radius (r) can be calculated by dividing the diameter by 2:
Radius = Diameter / 2 = 3.89 cm / 2 = 1.945 cm
Now, we can calculate the volume of the sphere:
Volume of the ball = (4/3) * π * (1.945 cm)^3
Next, we need to convert the volume from cubic centimeters (cm3) to liters (L). Since 1 L = 1000 cm3, we can divide the volume by 1000 to convert it to liters.
Volume of the ball in liters = Volume of the ball in cm3 / 1000
Once we have the volume of the ball in liters, we can calculate the weight of the displaced water using the density of water, which is approximately 1 g/cm3.
Weight of displaced water = Volume of the ball in liters * density of water
Finally, the force required to hold the ball completely submerged under water is equal to the weight of the displaced water, which is the buoyant force.
Therefore, the force required to hold the ball completely submerged under water can be calculated using the above steps.