V-belt

Calculate the length of the V-belt when the diameter of the pulleys is:

D1 = 600 mm
D2 = 120 mm
d = 480 mm (distance between pulley axes)

Correct answer:

l =  2214 mm

Step-by-step explanation:

$$\begin{gathered} d_1=600\text{mm} \\ {d}_2=120\text{mm} \\ d=480\text{mm} \\ {r}_1=d_1/2=600/2=300\text{mm} \\ {r}_2=d_2/2=120/2=60\text{mm} \\ a=r_1-r_2=300-60=240\text{mm} \\ \phi =\arcsin (a/d)=\arcsin (240/480)\doteq 0.5236\text{rad} \\ \alpha =\phi \to \text{°}=\phi \cdot \frac{180}{\pi }\text{°}=0.5236\cdot \frac{180}{\pi }\text{°}=30\text{°} \\ {A}_1=\pi +2\cdot \phi =3.1416+2\cdot 0.5236\doteq 4.1888\text{rad} \\ {A}_2=\pi -2\cdot \phi =3.1416-2\cdot 0.5236\doteq 2.0944\text{rad} \\ {l}_1=r_1\cdot A_1=300\cdot 4.1888\doteq 1256.6371\text{mm} \\ {l}_2=r_2\cdot A_2=60\cdot 2.0944\doteq 125.6637\text{mm} \\ b=\sqrt{d^2-a^2}=\sqrt{480^2-240^2}=240\sqrt{3}\text{mm}\doteq 415.6922\text{mm} \\ l=l_1+l_2+2\cdot b=1256.6371+125.6637+2\cdot 415.6922=2214\text{mm} \end{gathered}$$

Try another example