An electric motor runs at a constant rotational frequency of 3000 r/min and has the load of
75 Nm and a 5,0 Nm extra load of friction and gear. The efficiency of the motor is 78%. a)
What is the input power of the motor? b) What is the work done by the motor in 24 hours
and c) what is the total electrical energy needed in 24 hours?
To solve this problem, we'll need to use the formulas for power, work, and energy, and we'll also need to make a few assumptions.
a) To find the input power of the motor, we can use the formula: Power = Torque x Angular Velocity.
Since the load on the motor is given as 75 Nm and there is an additional load of 5.0 Nm due to friction and gear, the total torque is 75 Nm + 5.0 Nm = 80 Nm.
The rotational frequency of the motor is given as 3000 r/min, which is equivalent to (3000/60) Hz = 50 Hz.
To convert rotational frequency to angular velocity, we use the formula: Angular Velocity = 2π x Frequency.
Plugging in the values, we get: Angular Velocity = 2π x 50 = 100π rad/s.
Now we can calculate the input power: Power = Torque x Angular Velocity = 80 Nm x 100π rad/s.
b) To find the work done by the motor in 24 hours, we use the formula: Work = Power x Time.
Since the rotational frequency and torque remain constant, the power output of the motor remains constant as well. Therefore, the work done by the motor in 24 hours is simply the input power multiplied by the duration in seconds: Work = Power x (24 x 60 x 60) s.
c) To find the total electrical energy needed in 24 hours, we first need to calculate the energy consumed by the motor. The efficiency of the motor is given as 78%, which means that only 78% of the input power is converted into mechanical power. Therefore, the electrical energy needed is calculated as: Electrical Energy = Work / Efficiency.
Now that we understand the method, let's perform the calculations:
a) Input Power = 80 Nm x 100π rad/s = 8000π W (approximately 25133 W).
b) Work = 8000π W x (24 x 60 x 60) s = 69120000π J (approximately 217428045 J).
c) Electrical Energy = Work / Efficiency = 69120000π J / 0.78 = 88615385.2π J (approximately 278934481 J).
Therefore, the input power of the motor is approximately 25133 W, the work done by the motor in 24 hours is approximately 217428045 J, and the total electrical energy needed in 24 hours is approximately 278934481 J.