A ball of circumference 74.2 cm and mass 605 g is forced to the bottom of a swimming pool and then released. After initially accelerating upward, it rises at a constant velocity. (The density of water is 103 kg/m3.)
(a) Calculate the buoyant force on the ball. (Indicate the direction with the sign of your answer.)
(b) Calculate the drag force the ball experiences while it is moving upward at constant velocity. (Indicate the direction with the sign of your answer.)
Section
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To calculate the buoyant force on the ball, we need to use the equation:
Buoyant force = (density of fluid) * (volume of submerged part of the object) * (acceleration due to gravity)
First, let's calculate the volume of the submerged part of the ball. We know the circumference of the ball, so we can use the formula for the circumference of a circle:
Circumference = 2πr
Where r is the radius of the ball. Rearranging the formula, we find:
r = Circumference / (2π)
Substituting the given circumference of 74.2 cm into the equation, we have:
r = 74.2 cm / (2π)
Now, converting the units to meters, we get:
r = 0.742 m / (2π)
Simplifying, we find:
r ≈ 0.118 m
Now we can calculate the volume of the submerged part of the ball using the formula for the volume of a sphere:
Volume = (4/3)πr³
Substituting the calculated radius into the equation, we have:
Volume = (4/3)π(0.118 m)³
Simplifying, we find:
Volume ≈ 0.0039 m³
Next, we can calculate the buoyant force using the given density of water (103 kg/m³) and the acceleration due to gravity (9.8 m/s²):
Buoyant force = (density of fluid) * (volume of submerged part of the object) * (acceleration due to gravity)
Substituting the values, we have:
Buoyant force = (103 kg/m³) * (0.0039 m³) * (9.8 m/s²)
Calculating the numerical value, we get:
Buoyant force ≈ 3.9926 N
Since the ball is rising, the buoyant force acts in the upward direction. Therefore, the buoyant force is positive.
Now let's move on to calculating the drag force experienced by the ball while it is moving upward at a constant velocity.
The drag force experienced by an object moving through a fluid is given by the equation:
Drag force = 0.5 * (density of fluid) * (velocity of the object)² * (cross-sectional area of the object) * (drag coefficient)
Since the ball is rising at a constant velocity, the drag force is equal in magnitude but opposite in direction to the gravitational force acting on the ball, which is given by:
Gravitational force = (mass of the ball) * (acceleration due to gravity)
Substituting the given mass (605 g) into the equation, we have:
Gravitational force = (0.605 kg) * (9.8 m/s²)
Calculating the numerical value, we find:
Gravitational force ≈ 5.929 N
Since the ball is moving upward, the drag force acts in the downward direction. Therefore, the drag force is negative.
Now, to calculate the drag force, we need to determine the cross-sectional area of the ball. Since the ball is a sphere, the cross-sectional area is given by the equation:
Cross-sectional area = πr²
Substituting the calculated radius (0.118 m) into the equation, we have:
Cross-sectional area = π(0.118 m)²
Calculating the numerical value, we find:
Cross-sectional area ≈ 0.0439 m²
Now we can calculate the drag force using the given density of water (103 kg/m³), the velocity of the ball (which is constant), and a typical drag coefficient (which we'll assume to be 1):
Drag force = 0.5 * (density of fluid) * (velocity of the object)² * (cross-sectional area of the object) * (drag coefficient)
Since the ball is moving upward at a constant velocity, the velocity is zero. Substituting the values into the equation, we have:
Drag force = 0.5 * (103 kg/m³) * (0 m/s)² * (0.0439 m²) * (1)
Calculating the numerical value, we find:
Drag force = 0 N
Since the ball is moving upward, the drag force acts in the downward direction. Therefore, the drag force is negative.