A crane lifts a concrete hopper of mass 370 kg from the ground to the top of a 38 m building that is under construction. The hopper moves with a constant speed during the lift. The power rating of the crane motor is 8.2 kiloWatts.
How long does it take for the crane to lift the hopper to the top of the building?
(Work per hopper load)/(Time) = 8200 W
Time = M g H/P
about 17 seconds
To find out how long it takes for the crane to lift the hopper to the top of the building, we can use the concept of power.
Power is defined as the rate at which work is done, and it is given by the equation: Power = Work / Time.
We can rearrange this equation to solve for time: Time = Work / Power.
In this case, we need to calculate the work done by the crane to lift the hopper to the top of the building. The work done is equal to the change in potential energy of the hopper.
The potential energy of an object is given by the equation: Potential Energy = Mass * Gravity * Height.
In this case, the mass of the hopper is 370 kg, the height of the building is 38 m, and the acceleration due to gravity is approximately 9.8 m/s².
Plugging these values into the equation for potential energy, we get: Potential Energy = 370 kg * 9.8 m/s² * 38 m.
Now, we can calculate the work done by multiplying this potential energy by -1 (since the hopper is lifted against gravity): Work = -1 * Potential Energy.
Next, we convert the power rating of the crane motor from kilowatts to watts. Since 1 kilowatt is equal to 1000 watts, the power rating of the motor is 8.2 kilowatts * 1000 = 8200 watts.
Finally, we can substitute the values of work and power into the equation for time to find the answer: Time = Work / Power.
Let's calculate it step by step:
Potential Energy = 370 kg * 9.8 m/s² * 38 m = 138196 N·m
Work = -1 * Potential Energy = -138196 N·m
Power = 8200 watts
Time = Work / Power = -138196 N·m / 8200 watts.
However, since power is defined as work divided by time, a negative value for time does not make physical sense. Therefore, we can conclude that the crane is unable to lift the hopper with the given power rating.
In order to calculate the time it takes to lift the hopper to the top of the building with a constant speed, we would need to know the efficiency of the crane or provide additional information about the system.