Given Information:
- Motor speed: 1500 rpm
- Hysteresis loss: 180 W
- Eddy current loss: 70 W

Objective:
To find the motor speed at which the total loss will be one-fourth of the given losses.

Assumption:
Since the flux remains constant, we can assume that the magnetic field strength remains constant.

Explanation:
The losses in a motor can be categorized into two types: hysteresis loss and eddy current loss. These losses are caused by the magnetic properties of the motor's core and the circulating currents induced in the core, respectively.

Hysteresis Loss:
Hysteresis loss is caused by the magnetization and demagnetization of the magnetic materials in the motor's core as the magnetic field changes polarity. It is given by the equation:

Ph = kh * ƒ * B^1.6

where:
Ph = hysteresis loss
kh = hysteresis coefficient
ƒ = frequency (rpm/60)
B = magnetic field strength

From the given information, we know that the hysteresis loss is 180 W at 1500 rpm. Let's assume the hysteresis coefficient (kh) and magnetic field strength (B) remain constant. Therefore, we can write:

Ph1 = kh * (1500/60) * B^1.6 = 180 W

Eddy Current Loss:
Eddy current loss is caused by the circulating currents induced in the motor's core due to the varying magnetic field. It is given by the equation:

Pe = ke * ƒ^2 * B^2 * A^2 * d^2

where:
Pe = eddy current loss
ke = eddy current coefficient
ƒ = frequency (rpm/60)
B = magnetic field strength
A = area of the core
d = thickness of the core

From the given information, we know that the eddy current loss is 70 W at 1500 rpm. Let's assume the eddy current coefficient (ke), magnetic field strength (B), area of the core (A), and thickness of the core (d) remain constant. Therefore, we can write:

Pe1 = ke * (1500/60)^2 * B^2 * A^2 * d^2 = 70 W

Total Loss:
The total loss in the motor is the sum of hysteresis loss and eddy current loss. Therefore, the total loss (Pt) is given by:

Pt = Ph + Pe

Substituting the values of Ph1 and Pe1, we can write:

Pt1 = 180 + 70 = 250 W

Calculating the New Motor Speed:
We need to find the motor speed at which the total loss will be one-fourth of the given losses. Let's assume this speed as N rpm. The total loss at this speed (Pt2) will be:

Pt2 = 1/4 * Pt1 = 1/4 * 250 = 62.5 W

We can now substitute the values of Pt2, Ph, and Pe into the equations and solve for N:

Ph2 = kh * (N/60) * B^1.6
Pe