A uniform brass disk of radius R and mass M with a moment of inertia I about its cylindrical axis of symmetry is at a temperature T = 45 °C. Determine the fractional change in its moment of inertia if it is heated to a temperature of 125 °C. (The linear expansion coefficient for brass is 1.90 10-5 °C−1.)
I tried finding the change in volume and using that ratio, but it isn't correct. Im not sure what the best approach is. Thank you for any suggestions!
You have been given the linear expansion coefficient - find increase in radius R and then find the disk's MI.
ΔR=R•α•ΔT=R•1.9•10^-5•(125-45)=0.00152•R
R1=R+0.00152•R =R•1.001525
I=mR²/2
I1= mR1²/2= mR²•1.001525²/2=1.003• mR²/2=1.003•I
I1/I=1.003 or
I/I1=1/1.003=0.997
thank you!
To determine the fractional change in the moment of inertia of a uniform brass disk when it is heated from 45°C to 125°C, we need to consider the expansion of the disk due to the temperature change.
The change in moment of inertia is directly related to the change in mass and the change in the distribution of mass around the axis of rotation.
Here's how you can approach this problem:
1. Start by finding the change in linear dimensions of the disk. To do this, you need to use the linear expansion coefficient for brass (α = 1.90 × 10^(-5) °C^(-1)). The formula for linear expansion is:
ΔL = αLΔT
where ΔL is the change in length, α is the linear expansion coefficient, L is the original length, and ΔT is the change in temperature.
In this case, since we are dealing with a disk, the length L is equal to 2πR, where R is the radius of the disk.
So, the change in linear dimensions of the disk can be found using the formula:
ΔR = αRΔT
where ΔR is the change in radius.
2. Next, we need to find the change in volume of the disk. Since the disk is uniform, the density of the material remains constant. Therefore, the change in volume is equal to the change in mass divided by the density of the material (ρ):
ΔV = Δm/ρ
where ΔV is the change in volume and Δm is the change in mass.
The change in mass can be found using the formula:
Δm = ρπ((R+ΔR)^2 - R^2)((2π(R+ΔR))/2 - (2πR)/2)
Simplifying this expression yields:
Δm = ρπ(2RΔR+ΔR^2)
3. Now we can calculate the fractional change in moment of inertia. The moment of inertia of a uniform disk rotating about its cylindrical axis is given by the formula:
I = (1/2)MR^2
where M is the mass of the disk and R is the radius.
The change in moment of inertia is then given by:
ΔI = (1/2)(M+Δm)(R+ΔR)^2 - (1/2)MR^2
Finally, the fractional change in moment of inertia is calculated as:
Fractional change = (ΔI/I) = ΔI /[(1/2)MR^2]
You can plug in the values you have (R, M, α, ΔT) into the formulas above to find the fractional change in moment of inertia when the disk is heated.