A Styrofoam ball of radius 60 cm falls with a terminal velocity of 8.65 m/s. What is the mass of the ball?
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To determine the mass of the Styrofoam ball, we can use the formula for terminal velocity:
v = (2 * m * g) / (ρ * A * C),
where:
- v is the terminal velocity (8.65 m/s),
- m is the mass of the ball (what we want to find),
- g is the acceleration due to gravity (approximately 9.8 m/s^2),
- ρ is the density of the fluid (in this case, air),
- A is the cross-sectional area of the ball,
- C is the drag coefficient.
Since the ball is falling in air, we can assume that ρ (density of air) is constant and equal to about 1.2 kg/m^3. The cross-sectional area of a sphere can be calculated using the formula A = π * r^2, where r is the radius.
Given that the radius of the ball is 60 cm (or 0.6 m), we can find the mass by rearranging the formula:
m = (v * ρ * A * C) / (2 * g).
Calculating the values:
A = π * r^2 = π * (0.6 m)^2 = 1.13 m^2,
C is a dimensionless quantity that depends on the shape and properties of the object. Since the ball is made of Styrofoam, which is a relatively light and porous material, let's assume a typical value of C = 0.5.
Substituting the values into the formula:
m = (8.65 m/s * 1.2 kg/m^3 * 1.13 m^2 * 0.5) / (2 * 9.8 m/s^2).
Simplifying the equation:
m = 3.92 kg.
Therefore, the mass of the Styrofoam ball is approximately 3.92 kg.