Answer:
The larger gear will rotate through 156°
Step-by-step explanation:
Arc Length
The arc length S of an angle θ on a circle of radius r is:
[tex]S = \theta r[/tex]
Where θ is expressed in radians.
The smaller gear of r1=3.7 cm drives a larger gear of r2=7.1 cm. The smaller gear rotates through an angle of θ1=300°.
Convert the angle to radians:
[tex]\displaystyle \theta_1=300*\frac{\pi}{180}=\frac{5\pi}{3}[/tex]
The arc length of the smaller gear is:
[tex]\displaystyle S_1=\frac{5\pi}{3}\cdot 3.7[/tex]
[tex]\displaystyle S_1=\frac{18.5\pi}{3}[/tex]
The larger gear rotates the same arc length, so:
[tex]\displaystyle S_2=\frac{18.5\pi}{3}[/tex]
[tex]\displaystyle \theta_2\cdot r_2=\frac{18.5\pi}{3}[/tex]
Solving for θ2:
[tex]\displaystyle \theta_2=\frac{18.5\pi}{3r_2}[/tex]
[tex]\displaystyle \theta_2=\frac{18.5\pi}{3*7.1}[/tex]
[tex]\theta_2=2.73\ radians[/tex]
[tex]\displaystyle \theta_2=2.73*\frac{180}{\pi}=156[/tex]
The larger gear will rotate through 156°
