A mixing beater consists of three thin rods, each 10.9 cm long. The rods diverge from a central hub, separated from each other by 120°, and all turn in the same plane. A ball is attached to the end of each rod. Each ball has cross-sectional area 3.70 cm2 and is so shaped that it has a drag coefficient of 0.640. The drag force on each ball is

R = 1/2 D ρ A v^2 where D is the drag coefficient, ρ the density of the fluid, A the cross-sectional area, and v the speed of the object moving through the fluid. a) Calculate the power input required to spin the beater at 1000 rev/min in water. b)The beater is taken out of the water and held in air. If the input power remains the same (it wouldn't, but if it did), what would be the new rotation speed?

To calculate the power input required to spin the beater at 1000 rev/min in water, we need to use the equation for power:

P = F * v

where P is the power, F is the force, and v is the velocity.

a) To find the force, we need to calculate the drag force on each ball. The equation for drag force is given as:

R = 1/2 * D * ρ * A * v^2

where R is the drag force, D is the drag coefficient, ρ is the density of the fluid, A is the cross-sectional area of the ball, and v is the speed of the ball moving through the fluid.

First, let's calculate the drag force on each ball:

R = 1/2 * 0.640 * ρ * 3.70 cm^2 * v^2

Next, let's calculate the total drag force on all three balls:

Total R = 3 * R

Since the balls are attached to the ends of the rods and all three turn in the same plane, the total drag force on the beater will be three times the drag force on each ball.

Now, we will calculate the power input required to spin the beater at 1000 rev/min, assuming the input power is equal to the power output required to overcome the drag force.

P = Total R * v

Since the beater is spinning at 1000 rev/min, we need to convert this to rad/s:

ω = (1000 rev/min) * (2π rad/rev) / (60 s/min)

Now, let's calculate the power input:

P = (3 * R) * ω

b) To find the new rotation speed when the beater is taken out of the water and held in air, we can assume that the power input remains the same. Therefore, the power input equation can still be used:

P = (3 * R) * ω

However, since the beater is now in air instead of water, the drag force will be different. In this case, the density of air (ρ_air) and the drag coefficient of the ball in air (D_air) will be used in the equation.

We can rearrange the power equation to find the new rotation speed:

ω_new = P / (3 * R_air)

where R_air is the drag force on each ball in air.

Now you have all the steps needed to calculate the power input and the new rotation speed.

a) To calculate the power input required to spin the beater at 1000 rev/min in water, we need to determine the drag force and then calculate the power based on the drag force and the angular velocity.

1. Determine the drag force on one ball:
- Drag force equation: R = 1/2 D ρ A v^2
- Given values: D = 0.640 (drag coefficient), ρ = density of water (approx. 1000 kg/m^3), A = 3.70 cm^2 = 3.70 * 10^-4 m^2, v = velocity of the ball
- Convert velocity from revolutions per minute (rev/min) to meters per second (m/s):
- 1000 rev/min = (1000 rev/min) * (2π rad/rev) * (1 min/60 s) ≈ 104.72 rad/s
- Substitute the values into the drag force equation:
R = 1/2 * 0.640 * 1000 * 3.70 * 10^-4 * (104.72)^2

2. Calculate the total drag force on all three balls:
- Since there are three rods and each has a ball, the total drag force will be three times the drag force calculated in step 1.

3. Calculate the power input:
- Power (P) is the product of angular velocity (ω) and torque (τ):
P = ω * τ
- Torque is the product of the drag force (R) and the perpendicular distance (r) from the axis of rotation to the point of application of the force (here, the length of the rod).
- Convert the length of the rod from centimeters to meters:
- Length of the rod = 10.9 cm = 10.9 * 10^-2 m
- Substitute the values into the power equation:
P = (104.72 rad/s) * (3 * R * 10.9 * 10^-2)

b) To calculate the new rotation speed when the beater is taken out of the water and held in air (assuming the input power remains the same), we can use the conservation of energy principle.

1. In the water, the power input is used to overcome the drag force and maintain the rotation speed. In air, the power input will still be the same, but there will be no drag force.

2. Since there is no drag force in air, all the power input will be used to increase the rotational kinetic energy:

Power input = Change in rotational kinetic energy / Time

3. The rotational kinetic energy is given by:

Rotational kinetic energy = 1/2 * Moment of inertia * (angular velocity)^2

4. If the moment of inertia remains the same when the beater is taken out of the water (which is a reasonable assumption), then the change in rotational kinetic energy will be:

Change in rotational kinetic energy = 1/2 * Moment of inertia * ((new angular velocity)^2 - (original angular velocity)^2)

5. Since the power input remains the same, we can set the equations for power input equal to the equation for the change in rotational kinetic energy and solve for the new angular velocity.

test