What Is the Final Temperature of a Heated Copper Ball?

[FishieKissie06]

A copper ball with a radius of 1.4cm is heated until its diameter has increased by 0.22mm. Assuming a room temperature of 22*C, find the final temperature of the ball.

well what i did was i used linear expansion for this and i didnt get it right. i used (delta)L=(alpha)initial L*(delta)Temperature...solving for temperature

 

[SoccaCrazy24]

Well in order to do this... you must use volume expansion since it is not a rod gaining length... so you must use the equation... (delta)V = 3*alpha*V*(delta)T and for (delta)V you can substitute in V1-V2
and remember that you must have it in raidus and not diameter when finding (delta)V so... since the radius then increases by (d/2) you have to remember to convert to meters for both of the radius...

 

[quicknote]

I would like to clarify that the linear expansion equation you used is correct. However, it is important to make sure that all the units are consistent. In this case, the units for length should be in meters and the units for temperature should be in Kelvin. Additionally, it is necessary to use the correct value for the coefficient of linear expansion for copper, which is 1.7 x 10^-5 m/mK. With these adjustments, the final temperature of the copper ball can be calculated as follows:

First, convert the initial diameter increase from millimeters to meters:
Delta L = 0.22mm = 0.00022m

Next, calculate the initial length of the copper ball:
Initial L = 2 * radius = 2 * 0.014m = 0.028m

Then, plug in the values into the linear expansion equation:
Delta L = (alpha) * initial L * (delta)T
0.00022m = (1.7 x 10^-5 m/mK) * 0.028m * (delta)T

Solving for (delta)T:
(delta)T = 0.00022m / ((1.7 x 10^-5 m/mK) * 0.028m)
(delta)T = 0.914 K

Finally, to find the final temperature, add this value to the initial room temperature of 22°C, which is equivalent to 295 K:
Final temperature = 295 K + 0.914 K = 295.914 K

Therefore, the final temperature of the copper ball is approximately 295.9 K or 22.8°C. It is important to note that this is an approximation as the linear expansion equation assumes a uniform increase in length, which may not be the case for the copper ball. Other factors such as the specific heat capacity of copper and heat transfer to the surroundings may also affect the final temperature. Further experiments and calculations may be needed for a more accurate determination of the final temperature.