Calculate the force required to pull a copper ball of radius 3.00 cm upward through a fluid at the constant speed 7.00 cm/s. Take the drag force to be proportional to the speed, with proportionality constant 0.950 kg/s. Ignore the buoyant force.
Required force
F = M g + k V
k = 0.950 kg/s
M = (copper density)*(4/3) pi R^3
(make sure it is in kg)
g = 9.8 m/s^2
V = 0.07 m/s
R = 0.03 m
Do the numbers
To calculate the force required to pull the copper ball upward through a fluid at a constant speed, we need to consider two forces: the gravitational force and the drag force.
1. Gravitational force:
The gravitational force acting on the copper ball is given by the equation:
F_gravity = m * g
Where m is the mass of the ball and g is the acceleration due to gravity (approximately 9.8 m/s^2).
To find the mass of the copper ball, we can use the formula:
m = rho * V
Where rho is the density of copper and V is the volume of the ball.
Since the ball is a sphere, we can calculate its volume using the formula:
V = (4/3) * pi * r^3
Where r is the radius of the ball.
Substituting the values given:
r = 3.00 cm = 0.03 m
rho (density of copper) = 8,960 kg/m^3
V = (4/3) * pi * (0.03 m)^3
V ≈ 0.0001130973 m^3
Now, we can find the mass of the copper ball:
m ≈ 8,960 kg/m^3 * 0.0001130973 m^3
m ≈ 1.014 kg
Finally, we can calculate the gravitational force:
F_gravity ≈ 1.014 kg * 9.8 m/s^2
F_gravity ≈ 9.95052 N
2. Drag force:
The drag force acting on the copper ball is given by the equation:
F_drag = k * v
Where k is the proportionality constant and v is the speed of the ball.
Using the given values:
k = 0.950 kg/s
v = 7.00 cm/s = 0.07 m/s
Now we can calculate the drag force:
F_drag = 0.950 kg/s * 0.07 m/s
F_drag ≈ 0.0665 N
Since the ball is moving at a constant speed, the drag force will be equal to the force required to pull the ball upward, which is what we want to find.
Therefore, the force required to pull the copper ball upward through the fluid at a constant speed of 7.00 cm/s is approximately 0.0665 N.