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A 13.0 m long conveyor belt, inclined at 30.0°, is used to transport bundles of newspapers from the mailroom up to the cargo bay to be loaded on to delivery trucks. Each newspaper has a mass of 1.0 kg, and there are 23 newspapers per bundle. Determine the power that the conveyor develops if it delivers 15 bundles per minute.
Enter a number.1 W
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To determine the power that the conveyor develops, we can use the formula:
Power (P) = Work (W) / Time (t)
To find the work done by the conveyor, we need to calculate the force exerted on each newspaper bundle and the distance it is moved.
First, let's find the force exerted on each newspaper bundle. We know that the mass of each newspaper (m) is 1.0 kg and there are 23 newspapers per bundle. Therefore, the total mass of each bundle (M) is:
M = m * number of newspapers per bundle
M = 1.0 kg * 23
M = 23.0 kg
Next, we need to find the force exerted on the bundle in the direction of the inclined conveyor. This force is the weight of the bundle, which can be calculated using the formula:
Force (F) = Mass (M) * Acceleration due to gravity (g)
The acceleration due to gravity is approximately 9.8 m/s^2. Therefore:
F = 23.0 kg * 9.8 m/s^2
F = 225.4 N
Now let's calculate the distance each bundle is moved. Since the conveyor belt is inclined at an angle of 30.0°, the vertical distance (h) each bundle is lifted can be found using the formula:
h = conveyor belt length * sin(angle)
h = 13.0 m * sin(30.0°)
h = 6.5 m
Finally, we can calculate the work done by the conveyor using the formula:
Work (W) = Force (F) * Distance (d)
Since the distance moved is the same as the vertical distance lifted, we have:
W = 225.4 N * 6.5 m
W = 1464.1 J (joules)
Now we can calculate the power using the given time of 15 bundles per minute. To convert this to seconds:
t = 15 bundles / 60 seconds
t = 0.25 bundles per second
Now we can substitute the values into the power formula:
P = W / t
P = 1464.1 J / 0.25 bundles per second
P = 5856.4 J/s (joules per second), which is equivalent to 5856.4 watts (W)
Therefore, the power that the conveyor develops is 5856.4 W.