Q: If the radius of a circle is $$0.327 m $$and the central angle is $$132.6^\circ$$, what is the length of the arc subtended by this angle? (Use $$\pi \approx 3.14)$$

$$0.327 \times \frac{132.6}{360} \times 2\pi \approx 0.755 m$$

Q: Convert $$132.6^\circ to $$radians.

$$132.6 \times \frac{\pi}{180} \approx 2.314$$\text{ radians}$

Q: If a gear with a radius of $$3.6 in. $$completes one full revolution, what is the distance traveled by a point on its edge?

$$2\pi \times 3.6 \approx 22.6$$\text{ in.}$

Q: A gear with radius $$5.4 in. $$rotates at $$120$$ rpm. What is the linear speed of a point on its edge? (Use $$\pi \approx 3.14)$$

$$120 \times 2\pi \times 5.4 \approx 4,073$$\text{ in./min}$

Q: If a gear with radius $$3.6 in. $$meshes with a gear of radius $$5.4 in$$., and the smaller gear rotates $$10$$ times, how many times does the larger gear rotate?

$$\frac{3.6}{5.4} \times 10 \approx 6.67$$ times

Q: What is the formula for converting degrees to radians?

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Q: A circle has a radius $$r. If $$the central angle is $$\theta$$ radians, what is the formula for the area of the sector?

$$A = \frac{1}{2} r^2$ \theta$$

Q: If the radius of a circle is $$0.327 m $$and the central angle is $$132.6^\circ$$, what is the area of the sector? (Use $$\pi \approx 3.14)$$

$$\frac{132.6}{360} \times \pi \times (0.327)^2$ \approx 0.124 m$^2$$

Question 1

What is the definition of a radian in terms of the radius and arc length of a circle?

Accepted Answer

A radian is the angle subtended at the center of a circle by an arc equal to the radius.

Question 2

Given a circle with radius $$r$$, what is the formula for the circumference $$C of $$the circle?

Accepted Answer

$$C = 2\pi r$$

Question 3

If the radius of a circle is $$0.327 m $$and the central angle is $$132.6^\circ$$, what is the length of the arc subtended by this angle? (Use $$\pi \approx 3.14)$$

Accepted Answer

$$0.327 	imes \frac{132.6}{360} 	imes 2\pi \approx 0.755 m$$

Question 4

Convert $$132.6^\circ to $$radians.

Accepted Answer

$$132.6 	imes \frac{\pi}{180} \approx 2.314$$	ext{ radians}$

Question 5

If a gear with a radius of $$3.6 in. $$meshes with a gear of radius $$5.4 in$$., what is the ratio of their angular velocities?

Accepted Answer

$$\frac{3.6}{5.4}$$

Question 6

A circle has a radius $$r. $$What is the length of the arc subtended by a central angle $$	heta (in $$radians)?

Accepted Answer

$$r	heta$$

Question 7

If a gear with a radius of $$3.6 in. $$completes one full revolution, what is the distance traveled by a point on its edge?

Accepted Answer

$$2\pi 	imes 3.6 \approx 22.6$$	ext{ in.}$

Question 8

A circle has a radius of $$r. $$What is the area $$A of $$the circle?

Accepted Answer

$$A = \pi r^2$

Question 9

If the radius of a circle is doubled, how does the circumference change?

Accepted Answer

The circumference is doubled.

Question 10

A gear with radius $$5.4 in. $$rotates at $$120$$ rpm. What is the linear speed of a point on its edge? (Use $$\pi \approx 3.14)$$

Accepted Answer

$$120 	imes 2\pi 	imes 5.4 \approx 4,073$$	ext{ in./min}$

Question 11

If the arc length of a circle is $$2r$$ and the radius is $$r$$, what is the central angle in radians?

Accepted Answer

$$	heta = 2$$

Question 12

A circular saw blade has a radius of $$0.327 m $$and rotates through an angle of $$132.6^\circ. $$What fraction of the total circumference does this arc represent?

Accepted Answer

$$\frac{132.6}{360}$$

Question 13

If a gear with radius $$3.6 in. $$meshes with a gear of radius $$5.4 in$$., and the smaller gear rotates $$10$$ times, how many times does the larger gear rotate?

Accepted Answer

$$\frac{3.6}{5.4} 	imes 10 \approx 6.67$$ times

Question 14

What is the formula for converting degrees to radians?

Accepted Answer

$$	ext{radians} = 	ext{degrees} 	imes \frac{\pi}{180}$$

Question 15

A circle has a radius $$r. If $$the central angle is $$	heta$$ radians, what is the formula for the area of the sector?

Accepted Answer

$$A = \frac{1}{2} r^2$ 	heta$$

Question 16

If a gear with radius $$5.4 in. $$rotates once, how many times must a gear with radius $$3.6 in. $$rotate to match the same linear distance?

Accepted Answer

$$\frac{5.4}{3.6} = 1.5$$ times

Question 17

A circle has a radius $$r. $$What is the relationship between the arc length $$s$$, radius $$r$$, and central angle $$	heta (in $$radians)?

Accepted Answer

$$s = r	heta$$

Question 18

If the radius of a circle is $$0.327 m $$and the central angle is $$132.6^\circ$$, what is the area of the sector? (Use $$\pi \approx 3.14)$$

Accepted Answer

$$\frac{132.6}{360} 	imes \pi 	imes (0.327)^2$ \approx 0.124 m$^2$$

Angles, Radian Measure, and Applications of Circular Motion

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