Calculating Length, Diameter and Volume Change of a Laser Rod

[Knfoster]

Homework Statement

The active element of a certain laser is an ordinary glass rod 18.7 cm long and 1.33 cm in diameter. If the temperature of the rod increases by 78.5oC, calculate its increase in length. Calculate its increase in diameter. Calculate its increase in volume.

Homework Equations

change in L=Li*alfa*change in T

The Attempt at a Solution

Length increased by 0.0132 cm
Diameter increased by 9.396E-4cm
I don't know how to get the increase in volume. Could someone please help? Thanks!

 

[hage567]

You can calculate the intial volume easily enough. Look up volume expansion in your book. It should be in there. You can also look at http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thexp2.html#c3

 

[nlightner]

I would approach this problem by first identifying the relevant equations and variables. From the given information, we know that the initial length (Li) is 18.7 cm and the initial diameter (Di) is 1.33 cm. We are also given the change in temperature (ΔT) as 78.5°C. The coefficient of thermal expansion (α) for glass is typically around 8.5 x 10^-6 m/m°C.

Based on these variables, we can use the equation ΔL = Li*α*ΔT to calculate the change in length. Plugging in the values, we get ΔL = (18.7 cm)*8.5 x 10^-6 m/m°C*(78.5°C) = 0.0132 cm. This matches the given answer for the increase in length.

To calculate the change in diameter, we can use a similar equation ΔD = Di*α*ΔT. Plugging in the values, we get ΔD = (1.33 cm)*8.5 x 10^-6 m/m°C*(78.5°C) = 9.396 x 10^-4 cm. This matches the given answer for the increase in diameter.

To calculate the change in volume, we can use the equation ΔV = Vi*α*ΔT, where Vi is the initial volume. The initial volume of the rod can be calculated using the formula for the volume of a cylinder, V = πr^2h, where r is the radius (half of the diameter) and h is the height (length) of the cylinder. Plugging in the initial values, we get Vi = π*(0.665 cm)^2*(18.7 cm) = 7.85 cm^3.

Now, we can plug in the values for Vi, α, and ΔT into the equation for ΔV to get ΔV = (7.85 cm^3)*8.5 x 10^-6 m/m°C*(78.5°C) = 0.0055 cm^3. This is the increase in volume of the glass rod.

In summary, the increase in length is 0.0132 cm, the increase in diameter is 9.396 x 10^-4 cm, and the increase in volume is 0.0055 cm^3. These calculations show the expected changes in