Hi! help me please, Thank you.
A spherical steel ball bearing has a diameter of 2.540 cm at 23°C.
(a) What is the diameter when its temperature is raised to 94°C?
cm
(b) What temperature change is required to increase its volume by 0.9%? I only need help on part b.Thank
The subject is mispelled, again. Pleases get it right.
Volume is increased 0.9% when linear dmensions are increaed 0.3%, when the relative chance is much less than 1, as is the case here.
Sure, I can help you with part (b).
To determine the temperature change required to increase the volume of the spherical steel ball bearing by 0.9%, we can use the concept of thermal expansion.
1. Start by finding the initial volume of the ball bearing. The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Where V is the volume and r is the radius.
2. Next, calculate the final volume of the ball bearing after the temperature change. Since the problem states that the volume is increased by 0.9%, we need to multiply the initial volume by 1 + 0.9%, which is equivalent to multiplying it by 1.009.
V_final = V_initial * 1.009
3. Now, find the initial and final temperatures in Celsius. Let's say the initial temperature is T_initial and the final temperature is T_final.
4. Convert the temperatures from Celsius to Kelvin by adding 273.15 to each temperature value. This is because temperature changes in the Kelvin scale are directly proportional to the volume changes.
T_initial_Kelvin = T_initial + 273.15
T_final_Kelvin = T_final + 273.15
5. Apply the volume expansion formula:
(V_final / V_initial) = (T_final_Kelvin / T_initial_Kelvin)
Substitute the values to solve for the unknown T_final:
1.009 = (T_final_Kelvin / T_initial_Kelvin)
6. Rearrange the formula to solve for T_final:
T_final_Kelvin = 1.009 * T_initial_Kelvin
7. Convert the final temperature back from Kelvin to Celsius by subtracting 273.15.
T_final = T_final_Kelvin - 273.15
Now, you can plug in the values for the initial temperature (T_initial) to solve for the required temperature change (T_final - T_initial) in order to increase the volume by 0.9%.
I hope that helps! Let me know if you have any further questions.