At its normal operating speed, an electric fan motor draws only 10.9% of the current it draws when it just begins to turn the fan blade. The fan is plugged into a 120.0-V socket. What back emf does the motor generate at its normal operating speed?
The current when it "just begins to turn the blade" is a starting surge current. There is no back emf (Vb) at that time. Let R be the resistance of the motor coils.
The starting surge current is 120V/R.
The current at normal operating speed is (120 - Vb)/R
(120 - Vb)/R = 0.109*(120/R)
Vb = 120 - 13.08 = 106.9 V
Well, it seems the electric fan motor is quite the energy saver! It only draws 10.9% of the current it needs to get the fan blade turning. That's like going from a sprint to a leisurely stroll!
Now, to find the back emf the motor generates at its normal operating speed, we can use a little bit of math. Since power is equal to voltage times current (P = V x I), and power is also equal to back emf times current (P = Emf x I), we can make an equation.
At full force, the fan motor draws full current (100%). But at normal operating speed, it only draws 10.9% of that current. So, let's call the full current I0, and the normal operating speed current In.
Now, we can set up the equation:
120.0V x I0 = Emf x I0
Since I0 = 100%, and In = 10.9%, we can rewrite the equation as:
120.0V x 100% = Emf x 10.9%
Simplifying a bit further:
120.0V = Emf x 10.9%
And finally, solving for Emf:
Emf = 120.0V / 10.9%
Emf ≈ 1,100.92V
So, it looks like at its normal operating speed, the electric fan motor generates a back emf of around 1,100.92 volts. That's one electrifying fan!
To find the back emf generated by the motor at its normal operating speed, we need to use the concept of conservation of energy.
1. We know that the power input to the motor is equal to the power output.
2. The power input to the motor can be calculated using the formula: P = V * I, where P is power, V is voltage, and I is current.
3. The power output of the motor is calculated as the product of the back emf (E) and the current drawn by the motor at its normal operating speed (I_n).
4. Since the current drawn at the beginning is 100% (I_0 = 100%) and the current drawn at normal operating speed is 10.9% (I_n = 10.9%), we can calculate I_n as a percentage of I_0.
I_n = (10.9/100) * I_0
5. Now we can solve for back emf (E) by equating the power input and output.
V * I_0 = E * I_n
6. Substituting the values and solving for E:
120.0 * I_0 = E * ((10.9/100) * I_0)
E = (120.0 * I_0) / ((10.9/100) * I_0)
7. Simplifying the equation:
E = 120.0 / (10.9/100)
E = 120.0 / 0.109
E ≈ 1100 V
Therefore, the back emf generated by the motor at its normal operating speed is approximately 1100 V.
To find the back emf generated by the electric fan motor at its normal operating speed, we need to use Ohm's Law and the concept of power.
First, let's understand the relationship between back emf, current, and voltage:
1. Ohm's Law states that voltage (V) is equal to the product of current (I) and resistance (R): V = I * R.
2. In an electric motor, the back emf (E) is induced due to the rotation of the fan blade. The back emf opposes the flow of current, which is why it is often referred to as the counter-emf. Therefore, the back emf can be represented as E = I * R_back, where R_back is the effective resistance caused by the back emf.
Now, let's use the provided information to solve the problem:
1. We are given that the motor draws only 10.9% of the current it draws when starting. Let's assume the starting current is I_start. Therefore, the current at normal operating speed can be represented as I_normal = 0.109 * I_start.
2. The fan is plugged into a 120.0-V socket, which gives us the value of voltage (V) in Ohm's Law.
3. We need to determine the back emf (E) generated at normal operating speed. Since back emf opposes the flow of current, we can consider the back emf as a voltage drop, assuming the same effective resistance (R_back) for simplicity.
Plugging all the information into Ohm's Law (V = I * R), we get:
V = I_normal * R_back
120.0 V = (0.109 * I_start) * R_back
Now, note that we don't know the starting current (I_start) or the effective resistance (R_back). However, we have enough information to find the ratio between the normal operating speed current and the starting current:
I_normal = 0.109 * I_start
Dividing both sides of the equation by I_start, we have:
I_normal / I_start = 0.109
Now, consider the ratio of the back emf to the voltage:
E / V = R_back / R
Since the effective resistance caused by the back emf should remain constant, we can equate the ratio to the one we found above:
E / V = 0.109
Now, rearrange the equation to find the back emf (E):
E = V * 0.109
E = 120.0 V * 0.109
E = 13.08 V
Therefore, the back emf generated by the electric fan motor at its normal operating speed is approximately 13.08 V.