Change in radius over time for a spherical ball formula

In summary, the conversation discusses the use of symbols in algebra and how to differentiate using the product rule. It also introduces the ideal gas law and its application to finding the rate of change for volume and temperature. The conversation concludes with a suggestion to replace certain terms with their respective values and a reminder that N equals to n times N_A.
  • #1
aspodkfpo
148
5
Homework Statement
https://www.asi.edu.au/wp-content/uploads/2015/08/NQE_2008_Physics.pdf
Pg 7
Q 12 E)
Relevant Equations
v^r = c dT/dt
P = rG
PV = NkT
1598193191413.png

Algebra in this answer does not seem to flow right. Firstly, the 16, secondly the n term.
Can someone explain or show me the right answer?
 
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  • #2
Since big ##V## is used for voltage, denote the volume with little ##v##. You know ##v = \frac{4}{3}\pi r^3##, so $$\frac{dv}{dt} = 4\pi r^2 \frac{dr}{dt}$$Now you will require the ideal gas law,$$pv = (rG)v = NkT$$ Use the product rule when differentiating both sides w.r.t. ##t##,$$Gr\frac{dv}{dt} + Gv\frac{dr}{dt} = Nk \frac{dT}{dt}$$What do you get if you replace ##v## with ##\frac{4}{3}\pi r^3##, and also replace ##\frac{dv}{dt}## with ##4\pi r^2 \frac{dr}{dt}##? Also, it is helpful to note that ##N = nN_A##.
 
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1. What is the formula for calculating the change in radius over time for a spherical ball?

The formula for calculating the change in radius over time for a spherical ball is: Δr/Δt = (3/2)αr, where Δr is the change in radius, Δt is the change in time, α is the coefficient of thermal expansion, and r is the initial radius of the ball.

2. How is the change in radius affected by the coefficient of thermal expansion?

The change in radius is directly proportional to the coefficient of thermal expansion. This means that as the coefficient of thermal expansion increases, the change in radius will also increase.

3. Can this formula be used for any type of spherical ball?

Yes, this formula can be used for any type of spherical ball as long as the coefficient of thermal expansion remains constant and the ball is made of a uniform material.

4. How does the change in radius over time affect the volume of the spherical ball?

The change in radius over time directly affects the volume of the spherical ball. As the radius increases, the volume also increases. Similarly, as the radius decreases, the volume decreases.

5. Is the change in radius over time constant for a spherical ball?

No, the change in radius over time is not constant for a spherical ball. It depends on the coefficient of thermal expansion and the initial radius of the ball. As time increases, the change in radius may also change due to external factors such as temperature and pressure.

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