Pls with explanatim
*)An iron tyre of diameter 50cm at 15°c is to be shrunk on to a wheel of diameter 50.35cm.To what temperature must the tyre be heated so that it will slip over the wheel with a radial gap of 0.5mm?(co-efficient of linear expansion of iron 0.0000121°c)
Well, well, well, looks like we've got a hot question here! So, we have an iron tyre that needs to be shrunk onto a wheel. To figure out the temperature requirement, we need to break out our trusty equations.
Now, we know that the tyre's original diameter is 50 cm and the wheel's diameter is 50.35 cm. We want to find the temperature at which the tyre will slip over the wheel with a radial gap of 0.5 mm.
The first thing we need to calculate is the change in diameter of the iron tyre. Since the tyre needs to fit a little loose on the wheel, we subtract the radial gap from the wheel's diameter. So, the change in diameter is 0.035 cm (50.35 cm - 50 cm - 0.05 cm).
Now, we can use the formula for linear expansion. The formula is ΔL = L0 * α * ΔT, where ΔL is the change in length, L0 is the original length, α is the coefficient of linear expansion, and ΔT is the change in temperature.
In our case, the change in length is the change in diameter, L0 is the original diameter, α is the coefficient of linear expansion, and we need to find ΔT.
Rearranging the formula, we have ΔT = ΔL / (L0 * α).
Plugging in the values, we get ΔT = 0.035 cm / (50 cm * 0.0000121 °C).
Calculating that, we find ΔT ≈ 57.85 °C.
So, to slip that iron tyre over the wheel with a radial gap of 0.5 mm, you'll need to heat it to approximately 57.85 °C. Just remember, don't burn yourself in the process!
Step 1: Calculate the change in temperature needed for the tyre to expand enough to slip over the wheel with the given radial gap.
To determine the change in temperature, we can use the equation for linear expansion:
ΔL = α * L0 * ΔT
Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the original length
ΔT is the change in temperature
In this case, we know the radial gap, which is the change in length. We can rearrange the formula to solve for the change in temperature:
ΔT = ΔL / (α * L0)
Step 2: Find the change in length and original length of the iron tyre.
The original length of the tyre is the circumference of the tyre at the starting temperature. Since the diameter is given as 50 cm, we can calculate the original length as:
L0 = π * d0
Where:
L0 is the original length
π is a mathematical constant approximately equal to 3.14159
d0 is the diameter of the tyre at 15°C
For the change in length, we can subtract the original length from the final circumference with the radial gap:
ΔL = (π * d1) - L0
Where:
ΔL is the change in length
π is a mathematical constant approximately equal to 3.14159
d1 is the diameter of the tyre at the desired temperature
Step 3: Calculate the final diameter of the tyre.
The final diameter of the tyre is the sum of the wheel's diameter and twice the radial gap:
d1 = d_wheel + 2 * gap
Where:
d1 is the diameter of the tyre at the desired temperature
d_wheel is the diameter of the wheel
gap is the radial gap
Step 4: Substitute the values into the equation and solve for the change in temperature.
ΔT = ΔL / (α * L0)
ΔT = ((π * d1) - L0) / (α * L0)
Finally, substitute the known values for the equation:
ΔT = ((π * (d_wheel + 2 * gap)) - (π * d0)) / (α * π * d0)
Simplify the equation and calculate the result.
To determine the temperature at which the iron tyre needs to be heated in order for it to slip over the wheel with a radial gap of 0.5mm, you can follow these steps:
Step 1: Find the change in diameter of the iron tyre due to the change in temperature.
- The initial diameter of the iron tyre is 50cm, and the final diameter is 50.35cm. Therefore, the change in diameter is (50.35 - 50) = 0.35cm.
Step 2: Convert the change in diameter from centimeters to meters.
- Since the coefficient of linear expansion of iron is given in meters, the change in diameter needs to be converted to meters. 1 cm = 0.01 meters. So, the change in diameter in meters is 0.35cm * 0.01m/cm = 0.0035m.
Step 3: Calculate the change in temperature required using the formula for linear expansion:
- The formula for linear expansion is ΔL = α * L * ΔT, where ΔL is the change in length/diameter, α is the coefficient of linear expansion, L is the initial length/diameter, and ΔT is the change in temperature.
- Rearrange the formula to solve for ΔT: ΔT = ΔL / (α * L)
- Plugging in the values, we have ΔT = 0.0035m / (0.0000121°C^(-1) * 50m)
- Simplifying the equation, ΔT = 57.024°C.
Therefore, the iron tyre needs to be heated to a temperature of approximately 57.024°C for it to slip over the wheel with a radial gap of 0.5mm.