Estimate the terminal speed of a wooden sphere (density 0.700 g/cm3) falling through air if its radius is 3.20 cm and its drag coefficient is 0.470.
To estimate the terminal speed of a wooden sphere falling through air, we can use the following equation:
v = sqrt((2 * m * g) / (ρ * A * Cd))
Where:
v is the terminal speed,
m is the mass of the object (which we can calculate from the given density and radius),
g is the acceleration due to gravity (approximately 9.81 m/s^2),
ρ is the density of the fluid (air in this case),
A is the cross-sectional area of the object, and
Cd is the drag coefficient.
Let's calculate the different components step by step:
1. Calculate the mass (m) of the wooden sphere:
Given density = 0.700 g/cm^3 and radius = 3.20 cm.
Density of the wooden sphere (ρ) = 0.700 g/cm^3.
Volume of the sphere (V) = (4/3) * π * r^3, where r is the radius.
Converting the volume to cm^3: V = (4/3) * π * (3.20 cm)^3
Mass = Density * Volume
2. Calculate the cross-sectional area (A) of the sphere:
The cross-sectional area of a sphere is given by the equation A = π * r^2.
3. Now, we can calculate the terminal speed using the equation:
v = sqrt((2 * m * g) / (ρ * A * Cd))
Let's perform the calculations step by step.
To estimate the terminal speed of a wooden sphere falling through air, we can use the concept of terminal velocity. Terminal velocity is the steady speed reached by an object when the drag force acting on it is equal to the gravitational force pulling it downward.
The equation for terminal velocity is given by:
Vt = sqrt((2 * m * g) / (ρ * A * Cd))
where:
Vt = Terminal velocity
m = Mass of the object
g = Acceleration due to gravity (9.8 m/s^2)
ρ = Density of the medium (air in this case)
A = Cross-sectional area of the object
Cd = Drag coefficient of the object
To calculate the terminal velocity, we need to determine the mass and cross-sectional area of the wooden sphere.
1. Calculate the mass (m):
The volume of the sphere can be calculated using the formula:
V = (4/3) * π * r^3
where:
V = Volume of the sphere
π = Pi (approximately 3.14159)
r = Radius of the sphere
Substituting the given values, we get:
V = (4/3) * π * (3.20 cm)^3
Next, we can calculate the mass using the density formula:
m = ρ * V
Substituting the given density, we have:
m = 0.700 g/cm^3 * V
Convert cm^3 to m^3 by dividing by 1000:
m = 0.700 g/cm^3 * (V / 1000)
2. Calculate the cross-sectional area (A):
The cross-sectional area of a sphere can be calculated using the formula:
A = π * r^2
Substituting the given radius, we have:
A = π * (3.20 cm)^2
3. Calculate the terminal velocity (Vt):
Now, we can substitute the calculated values into the terminal velocity equation:
Vt = sqrt((2 * m * g) / (ρ * A * Cd))
Substitute the values for m, g, ρ, A, and Cd to obtain the estimate of the terminal speed of the wooden sphere falling through air.