A leather ball of circumference 75.9 m and mass 557 g is forced to the bottom of a swimming pool and then released. After initially accelerating upward, it rises at a constant velocity.
a) Calculate the buoyant force on the ball.
b) Calculate the drag force the ball experiences while it is moving upward at constant velocity.
a) To calculate the buoyant force on the ball, we can use Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
In this case, the ball is completely submerged in the swimming pool, so it displaces a volume of water equal to its own volume. The volume of the ball can be calculated using its circumference. The formula for the volume of a sphere is given by:
V = (4/3) * π * r^3,
where V is the volume and r is the radius of the sphere. The circumference of the ball is related to its radius by the equation:
C = 2 * π * r,
where C is the circumference. Rearranging the equation, we can solve for the radius:
r = C / (2 * π).
Substituting the given circumference of 75.9 m into the equation, we find:
r = 75.9 m / (2 * 3.14) ≈ 12.067 m.
Now we can calculate the volume of the ball:
V = (4/3) * π * (12.067 m)^3 ≈ 7300.75 m^3.
The density of water is approximately 1000 kg/m^3. Since the mass of the ball is given in grams, we need to convert it to kilograms:
mass = 557 g = 557 g / 1000 ≈ 0.557 kg.
The buoyant force is equal to the weight of the water displaced by the ball:
buoyant force = weight of water displaced = density of water * volume of water displaced * gravitational acceleration = 1000 kg/m^3 * 7300.75 m^3 * 9.8 m/s^2 ≈ 7.19 x 10^7 N.
Therefore, the buoyant force on the ball is approximately 7.19 x 10^7 Newtons.
b) When the ball is rising at a constant velocity, the drag force acting on it is equal in magnitude and opposite in direction to the force of gravity. This means that the drag force is the same as the weight of the ball.
weight of the ball = mass of the ball * gravitational acceleration = 0.557 kg * 9.8 m/s^2 ≈ 5.46 N.
Hence, the drag force the ball experiences while moving upward at constant velocity is approximately 5.46 Newtons.
To answer these questions, we need to understand the physical principles at play. The buoyant force and drag force can be calculated using the following equations:
a) Buoyant force (Fb) = weight of the fluid displaced by the object = fluid density (ρf) * volume of object submerged (Vsubmerged) * gravitational acceleration (g)
b) Drag force (Fd) = drag coefficient (Cd) * reference area (Aref) * fluid density (ρf) * velocity of the object (V)^2 / 2
Now, let's solve these questions step by step:
a) To calculate the buoyant force, we need the density of the fluid (water) and the volume of the ball submerged.
- Density of water (ρf) is approximately 1000 kg/m^3.
- Volume of the ball submerged (Vsubmerged) can be calculated using the circumference of the ball.
Given that the circumference of the ball is 75.9 m, we can use the equation for the circumference of a circle (C = 2πr) to find the radius (r) of the ball:
C = 2πr
75.9 = 2πr
r = 75.9 / (2π)
Once we have the radius, we can calculate the volume of the ball using the formula for the volume of a sphere (V = 4/3 * π * r^3) and then multiply it by 2 (since the ball is submerged halfway).
Vsubmerged = 2 * (4/3 * π * r^3)
Now we can calculate the buoyant force:
Fb = ρf * Vsubmerged * g
Substitute the values you've found into the equation to get the answer.
b) To calculate the drag force, we need the drag coefficient, reference area, fluid density, and velocity of the ball.
- The drag coefficient (Cd) depends on the shape and surface properties of the ball. Without any further information, we can assume a value around 0.47 for a smooth sphere.
- The reference area (Aref) can be approximated by the cross-sectional area of the ball, given by Aref = π * r^2.
- The fluid density (ρf) is the same as in part a.
- The velocity of the ball (V) is constant, so we need to take the upward velocity value into account.
Plug in the values and calculate the drag force using the drag force formula.
These calculations will give you the values for both the buoyant force and the drag force on the ball.