A 10.0 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 22 newspapers per bundle. Determine the power that the conveyor develops if it delivers 15 bundles per minute.
22 kg /bundle
work for one bundle = m g h = 22*9.8*5
bundles/second = 15/60
so seconds/bundle = 4
power = energy/time = 22*9.8*5/4 Watts
To determine the power developed by the conveyor, we need to find the work done per unit time, which is equal to the power.
First, let's find the total mass of the bundles that are being transported:
Total mass = mass per newspaper * number of newspapers per bundle * number of bundles per minute
Total mass = 1.0 kg * 22 newspapers/bundle * 15 bundles/minute
Next, let's find the vertical height the bundles are being transported:
Vertical height = length of the conveyor belt * sine of the inclination angle
Vertical height = 10.0 m * sin(30.0°)
Now, let's find the work done to transport the bundles:
Work done = force * distance
Work done = mass * acceleration due to gravity * vertical height
The acceleration due to gravity is approximately 9.8 m/s^2.
Finally, let's calculate the power:
Power = work done / time
Let's calculate each step:
Step 1:
Total mass = 1.0 kg * 22 newspapers/bundle * 15 bundles/minute
Total mass = 330 kg/minute
Step 2:
Vertical height = 10.0 m * sin(30.0°)
Vertical height = 10.0 m * 0.5
Vertical height = 5.0 m
Step 3:
Work done = mass * acceleration due to gravity * vertical height
Work done = 330 kg/minute * 9.8 m/s^2 * 5.0 m
Step 4:
Power = work done / time
Power = (330 kg/minute * 9.8 m/s^2 * 5.0 m) / 60 s/minute
Now, plug in the numbers and calculate the final answer.
To determine the power that the conveyor belt develops, we need to find the rate at which it does work, which is equal to the product of the force applied and the velocity of the objects being moved.
First, let's find the force applied by the conveyor belt on each newspaper bundle. The force can be determined using the equation:
Force = mass * acceleration
Since the conveyor belt is inclined at an angle of 30 degrees, we need to find the component of the weight of each newspaper bundle along the direction of the incline. The weight can be calculated using:
Weight = mass * gravity
The component of the weight along the incline is given by:
Weight_parallel = Weight * sin(theta)
where theta is the angle of the incline.
Now, we can calculate the force applied by the conveyor belt:
Force = Weight_parallel = (mass * gravity) * sin(theta)
Next, we need to find the velocity of the objects being moved. The velocity can be determined using the equation:
Velocity = distance / time
In this case, the distance is equal to the length of the conveyor belt (10.0 m), and the time is equal to the rate at which the bundles are delivered (15 bundles per minute). We need to convert the time to seconds:
Velocity = 10.0 m / (15 bundles/minute * 60 seconds/minute)
Finally, we can calculate the power using the formula:
Power = Force * Velocity
Substituting the values we have calculated, we can find the power developed by the conveyor belt.