Question

two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. if the smaller gear rotates through an angle of 270°, through how many degrees will the larger gear rotate? (smaller gear radius: 3.5 cm, larger gear radius: 6.9 cm)

Explanation:

Step1: Recall arc length formula

The arc length \( s \) is given by \( s = r\theta \) (where \( \theta \) is in radians). For two gears meshing, the arc length they move through is the same. First, convert \( 270^\circ \) to radians: \( 270^\circ\times\frac{\pi}{180^\circ}=\frac{3\pi}{2} \) radians. The radius of the smaller gear \( r_1 = 3.5 \) cm, so the arc length \( s = r_1\theta_1=3.5\times\frac{3\pi}{2} \).

Step2: Find angle for larger gear

Let the radius of the larger gear be \( r_2 = 6.9 \) cm and its angle be \( \theta_2 \) (in radians). Since \( s = r_2\theta_2 \), we have \( \theta_2=\frac{s}{r_2}=\frac{3.5\times\frac{3\pi}{2}}{6.9} \). Then convert back to degrees: \( \theta_2\times\frac{180^\circ}{\pi}=\frac{3.5\times3\times180^\circ}{2\times6.9} \).

Step3: Calculate the value

First, calculate numerator: \( 3.5\times3\times180 = 3.5\times540 = 1890 \). Denominator: \( 2\times6.9 = 13.8 \). Then \( \frac{1890}{13.8}\approx136.96^\circ \).

Answer:

Approximately \( 137^\circ \) (or more precisely \( \frac{3.5\times270}{6.9}\approx136.96^\circ \))