In a single plate disk clutch, the outer diameter is twice the inner diameter. We need to find the ratio of torque transmitted in the uniform pressure theory to the uniform wear theory, assuming the same axial load in both cases.

Let's assume the inner diameter of the clutch is 'd' and the outer diameter is '2d'.

Uniform Pressure Theory:
In the uniform pressure theory, the pressure acting on the clutch is assumed to be constant throughout the contact area. This means that the pressure distribution is uniform across the clutch surface.

Uniform Wear Theory:
In the uniform wear theory, the wear of the clutch material is assumed to be uniform across the contact surface. This theory takes into account the varying pressure distribution across the clutch surface.

To find the ratio of torque transmitted in the two theories, we can use the equation:

T = μ * P * r * w

where T is the torque transmitted, μ is the coefficient of friction, P is the pressure acting on the clutch, r is the radius of the clutch, and w is the contact width.

Let's consider the same axial load in both cases, which means the pressure acting on the clutch will be the same.

Uniform Pressure Theory:
In this case, the pressure is constant throughout the clutch surface. Therefore, the pressure distribution is uniform. The radius of the clutch is given by r = d.

Uniform Wear Theory:
In this case, the pressure distribution is not uniform. The radius of the clutch varies from the inner diameter (d) to the outer diameter (2d). Therefore, the average radius can be considered as (d + 2d)/2 = 3d/2.

Now, let's calculate the ratio of torque transmitted in the two theories:

Ratio = (T_uniform pressure theory) / (T_uniform wear theory)

Using the equation T = μ * P * r * w, we can cancel out the common terms of μ and P:

Ratio = (r_uniform pressure theory * w_uniform pressure theory) / (r_uniform wear theory * w_uniform wear theory)

Substituting the values of r and w for both cases:

Ratio = (d * w_uniform pressure theory) / ((3d/2) * w_uniform wear theory)

Since we are assuming the same axial load in both cases, the contact width (w) will remain the same.

Ratio = (d * w) / ((3d/2) * w) = 2/3

Therefore, the correct answer is (A) 2/3.