Question:
A lathe has to have five spindle speeds. It is to cater for work ranging from 25mm to 200mm in diameter. Allowing for a cutting speed of 25m/min, find the spindle speeds using:
a) Arithmetic Progression
b) Geometric Progression
Answer:
a) Firstly utilise the formula for Feed Rate:
![](https://static.wixstatic.com/media/20c892_8cb76583417043bcbc20d30d3c606947~mv2.png/v1/fill/w_76,h_7,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_8cb76583417043bcbc20d30d3c606947~mv2.png)
Calculate spindle speed for the 1st and 5th term in the series as we have the diameters for these terms:
1st term: Diameter = 200mm,
![](https://static.wixstatic.com/media/20c892_8d26a473c29644969792f52d3c286f3d~mv2.png/v1/fill/w_77,h_6,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_8d26a473c29644969792f52d3c286f3d~mv2.png)
5th term: Diameter = 25mm,
![](https://static.wixstatic.com/media/20c892_28a6c6edcd6f429ca798fbf1febd8f24~mv2.png/v1/fill/w_76,h_5,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_28a6c6edcd6f429ca798fbf1febd8f24~mv2.png)
The formula for Arithmetic Progression is as follows. Use Arithmetic formula and re-arrange to find the common difference (d):
![](https://static.wixstatic.com/media/20c892_c709ad9ef3944fa3a261732ceda59440~mv2.png/v1/fill/w_75,h_13,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_c709ad9ef3944fa3a261732ceda59440~mv2.png)
Where,
a = First term
d = Common difference
n = nth term
Now use formula to find remaining spindle speeds:
![](https://static.wixstatic.com/media/20c892_f3ab88093a0f40b0878a851ab51bbd75~mv2.png/v1/fill/w_74,h_14,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_f3ab88093a0f40b0878a851ab51bbd75~mv2.png)
Therefore according to Arithmetic Progression the 1st to 5th spindle speeds are
[40, 110, 179, 249, 318 rev/min].
b) We know from answer (a) that the 1st and 5th spindle speeds are:
![](https://static.wixstatic.com/media/20c892_8d26a473c29644969792f52d3c286f3d~mv2.png/v1/fill/w_77,h_6,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_8d26a473c29644969792f52d3c286f3d~mv2.png)
![](https://static.wixstatic.com/media/20c892_28a6c6edcd6f429ca798fbf1febd8f24~mv2.png/v1/fill/w_76,h_5,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_28a6c6edcd6f429ca798fbf1febd8f24~mv2.png)
The formula for Geometric Progression is as follows.
![](https://static.wixstatic.com/media/20c892_dc0353b7ae354979af59137dfad815a5~mv2.png/v1/fill/w_75,h_4,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_dc0353b7ae354979af59137dfad815a5~mv2.png)
Where,
a = First term
r = Common ratio
n = nth term
Use Geometric formula and re-arrange the 5th term to find the common ratio (r):
![](https://static.wixstatic.com/media/20c892_22d216c14b254b228b75cd044343764f~mv2.png/v1/fill/w_75,h_22,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_22d216c14b254b228b75cd044343764f~mv2.png)
Now use Geometric formula to find the remaining spindle speeds:
![](https://static.wixstatic.com/media/20c892_dc0353b7ae354979af59137dfad815a5~mv2.png/v1/fill/w_75,h_4,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_dc0353b7ae354979af59137dfad815a5~mv2.png)
![](https://static.wixstatic.com/media/20c892_02f996ab6e714698af147b5407010d7e~mv2.png/v1/fill/w_77,h_15,al_c,q_85,usm_0.66_1.00_0.01,blur_2,enc_auto/20c892_02f996ab6e714698af147b5407010d7e~mv2.png)
Therefore according to Geometric Progression the 1st to 5th spindle speeds are
[40, 67, 113, 189, 318 rev/min].