An iron plate at 20°c has a hole of radius of 8.92mm in the centre, an iron rivet with radius of 8.95mm at 20°c, inserted into the hole. To what temperature the plate heated for the rivet to fit into the hole. Linear expansivity of iron=1.24×10^-5/k
To determine the temperature to which the plate must be heated for the rivet to fit into the hole, we can use the concept of thermal expansion.
The change in radius of the hole can be calculated using the equation:
ΔR = α × R × ΔT
Where ΔR is the change in radius, α is the linear expansivity of iron, R is the initial radius, and ΔT is the change in temperature.
The change in radius of the rivet can be calculated in the same manner.
Since the rivet needs to fit into the hole, the change in radius of the hole must equal the change in radius of the rivet. Therefore, we can set up the equation:
α × R1 × ΔT = α × R2 × ΔT
Where R1 is the initial radius of the hole, R2 is the initial radius of the rivet, and ΔT is the change in temperature.
Plugging in the given values:
(1.24×10^-5/k) × (8.92mm) × ΔT = (1.24×10^-5/k) × (8.95mm) × ΔT
Simplifying the equation:
8.92mm × ΔT = 8.95mm × ΔT
The units of mm and k cancel out, leaving:
8.92ΔT = 8.95ΔT
The change in temperature, ΔT, cancels out:
8.92 = 8.95
The answer is 8.92 = 8.95, which is not possible. This means that the rivet will never fit into the hole, regardless of temperature changes.
Therefore, there is no temperature to which the plate can be heated for the rivet to fit into the hole.
Do not heat the rivet.
My apologies for the confusion. If we assume that only the plate is heated and the rivet remains at its original temperature, we can calculate the temperature to which the plate needs to be heated for the rivet to fit into the hole.
The change in radius of the hole can still be calculated using the equation:
ΔR = α × R × ΔT
Where ΔR is the change in radius, α is the linear expansivity of iron, R is the initial radius, and ΔT is the change in temperature.
Since the rivet remains at its original temperature, the change in radius of the hole must equal the change in radius of the rivet. Therefore, we can set up the equation:
α × R1 × ΔT = α × R2 × 0
Where R1 is the initial radius of the hole, R2 is the initial radius of the rivet, and ΔT is the change in temperature, which is assumed to be zero for the rivet.
Simplifying the equation:
α × R1 × ΔT = 0
Since ΔT is zero for the rivet, the equation becomes:
0 = 0
The answer is 0 = 0, which is true.
Therefore, the plate does not need to be heated for the rivet to fit into the hole, as the change in temperature for the rivet is assumed to be zero.
delta R = 8.95 - 8.92 !!!!!
Good Grief
Apologies for the error in the response. You are right.
To find the change in radius of the hole, we can use the equation:
ΔR = α × R × ΔT
Where ΔR is the change in radius, α is the linear expansivity of iron, R is the initial radius, and ΔT is the change in temperature.
Given:
α = 1.24×10^-5/k
R = 8.92mm
ΔR = 8.95mm - 8.92mm = 0.03mm
Plugging in the values into the equation, we have:
0.03mm = (1.24×10^-5/k) × (8.92mm) × ΔT
Simplifying the equation:
ΔT = 0.03mm / [(1.24×10^-5/k) × (8.92mm)]
Calculating the value:
ΔT ≈ 17.91 k
Therefore, the plate needs to be heated by approximately 17.91 Kelvin (or degrees Celsius) for the rivet to fit into the hole.