A crane is lifting a 500kg payload straight up at a constant speed of 0.7ms-1.
(a)What is the power output of the crane. (ignoring losses)?
(b)If it takes the cran2 2 minutes to raise the payload to its final height, how far above the ground is this?
(c)If the cable were to break as the payload reaches its final height, how fast would it be traveling as it hit the ground?
Someone will be happy to critique your work. Priority is given to students who make some sort of an effort.
Im not sure what equations to use other than the equation for potential energy or the equation for work. That is about as far as I have come to solving the problem.
To calculate the power output of the crane, we need to use the formula:
Power = Force × Velocity
Since the crane is lifting the payload straight up at a constant speed, we know that the force is equal to the weight of the payload, which can be calculated by multiplying the mass of the payload (500 kg) by the acceleration due to gravity (9.8 m/s²):
Force = mass × acceleration due to gravity
Force = 500 kg × 9.8 m/s² = 4900 N
Now, we can substitute the force and the velocity (0.7 m/s) into the power formula:
Power = Force × Velocity
Power = 4900 N × 0.7 m/s = 3430 Watts (or 3.43 kilowatts)
So, the power output of the crane is 3430 Watts (or 3.43 kilowatts).
Next, let's calculate how far above the ground the payload is after 2 minutes. We can use the formula:
Distance = Velocity × Time
Velocity is already given (0.7 m/s), and we need to convert 2 minutes to seconds:
2 minutes = 2 × 60 seconds = 120 seconds
Now, substitute the values into the formula:
Distance = Velocity × Time
Distance = 0.7 m/s × 120 seconds = 84 meters
Therefore, the payload is 84 meters above the ground.
Lastly, let's calculate the speed of the payload just before it hits the ground if the cable were to break. To do this, we can use the equation of motion:
Final Velocity² = Initial Velocity² + 2 × Acceleration × Distance
Since the payload is initially at rest, the initial velocity is 0 m/s. The acceleration due to gravity (g) is -9.8 m/s² (negative because it acts downward). The distance above the ground is 84 meters.
Substitute the values into the equation:
Final Velocity² = 0² + 2 × (-9.8 m/s²) × 84 meters
Final Velocity² = -2 × 9.8 m/s² × 84 meters
Final Velocity² = -1610.4 m²/s²
Taking the square root of both sides to find the final velocity:
Final Velocity = √(-1610.4 m²/s²)
(Note: The square root of a negative number is imaginary and not physically meaningful in this context, so the payload won't hit the ground if the cable breaks when it reaches its final height.)
Therefore, if the cable were to break, the payload would not hit the ground. Instead, it would remain at its final height of 84 meters above the ground.