an iron of diameter 4 cm at 25 degree and a cylinder of diameter 4.002 cm is made of the same material at the same temperature. to what temperature the ring should be heated so that the cylinder passes through the ring easily.?

a) 41.6 degree Celsius
b) 66.6 degree Celsius
c) 32.6 degree Celsius
d)76.6 degree Celsius

please show working and thank you

(b)

Linear coefficient of thermal expansion
α=ΔL/L•Δt
Δt = 0.002/4•12•10^-6=41.66º
t=25+41.66=66.6ºC

To find the temperature to which the ring should be heated, we can use the principle of thermal expansion.

The formula for thermal expansion is given by:

ΔL = α * L * ΔT

Where:
ΔL = Change in length
α = Coefficient of linear expansion
L = Initial length
ΔT = Change in temperature

Since both the iron ring and the cylinder are made of the same material and have the same initial diameter, we can assume that the coefficient of linear expansion (α) is the same for both.

The initial length of the ring and the cylinder is equal to the circumference. The circumference of a circle is given by:

C = π * d

Where:
C = Circumference
π = Pi (approximately 3.14159)
d = Diameter of the circle

Let's assume the initial length of both the ring and the cylinder is L.

The change in length for the ring (ΔL_ring) is equal to the change in length for the cylinder (ΔL_cylinder). Hence, we can equate the two expressions:

ΔL_ring = ΔL_cylinder

α * L_ring * ΔT_ring = α * L_cylinder * ΔT_cylinder

The diameters of the ring and the cylinder can be related as:

Diameter_ring = Diameter_cylinder + ΔDiameter

Given that:
Diameter_ring = 4 cm
Diameter_cylinder = 4.002 cm

We can calculate ΔDiameter:

ΔDiameter = Diameter_ring - Diameter_cylinder

Plugging in the values:

ΔDiameter = 4 cm - 4.002 cm
ΔDiameter = -0.002 cm

Since the change in diameter is negative, that means the ring needs to expand to accommodate the cylinder. Hence, we can replace ΔDiameter with -ΔDiameter in the equation.

We can express the change in length as:

ΔL_ring = α * L_ring * ΔT_ring
ΔL_cylinder = α * L_cylinder * ΔT_cylinder

And the change in diameter as:

ΔDiameter = -α * L_ring * ΔT_ring

By substituting the values:

-0.002 cm = -α * L_ring * ΔT_ring

Let's solve for ΔT_ring:

ΔT_ring = -0.002 cm / (-α * L_ring)

Since α and L_ring are constants, ΔT_ring is inversely proportional to -ΔDiameter. We need to find the temperature change such that the difference in diameter becomes zero (i.e., ΔDiameter = 0).

Let's solve for ΔT_ring when ΔDiameter = 0:

0 = -α * L_ring * ΔT_ring

ΔT_ring = 0

Therefore, the temperature to which the ring should be heated so that the cylinder passes through easily is 0°C.

None of the given options (a) 41.6°C, b) 66.6°C, c) 32.6°C, d) 76.6°C) is correct.

To solve this problem, we need to consider the concept of thermal expansion and the difference in diameter between the iron ring and the cylinder.

Thermal expansion is the tendency of a substance to expand when heated. The change in length or volume of a material due to temperature change can be calculated using the formula:

ΔL = α * L * ΔT

Where:
ΔL is the change in length or diameter
α is the coefficient of linear expansion
L is the original length or diameter
ΔT is the change in temperature

In this case, we want to find the temperature at which the ring should be heated so that the cylinder can pass through it easily. Since the cylinder has a slightly larger diameter, it means that the ring needs to expand to accommodate the cylinder.

Let's calculate the change in diameter for both the iron ring and the cylinder:

ΔD_ring = α * D_ring * ΔT
ΔD_cylinder = α * D_cylinder * ΔT

Since both the ring and the cylinder are made of the same material and at the same temperature, their coefficients of linear expansion (α) will be the same. Therefore, we can set their change in diameters equal to each other:

ΔD_ring = ΔD_cylinder

α * D_ring * ΔT = α * D_cylinder * ΔT

Since α and ΔT are common factors on both sides, we can cancel them out:

D_ring = D_cylinder

Substituting the given values, we have:

4 cm = 4.002 cm

Solving this equation, we find that D_ring = D_cylinder at the current temperature.

Therefore, the correct answer is that the ring does not need to be heated. The cylinder already passes through the ring easily at the given temperature.