Deck 16: Radian Measure, Arc Length, and Circular Motion

Question 1

Convert to radians: $83.2^{\circ}$

Answer 1

$1.45 \mathrm{rad}$

Question 2

Convert to radians: 1.92 rev

Answer 2

$12.1 \mathrm{rad}$

Question 3

Convert to radians: $37.8^{\circ}$

Answer 3

$0.660 \mathrm{rad}$

Question 4

Convert to radians: 7.40 rev

Question 5

Convert to radians in terms of $\pi: 120^{\circ}$

Question 6

Convert to radians in terms of $\pi: 342^{\circ}$

Question 7

Convert to radians in terms of $\pi: 63^{\circ}$

Question 8

Convert to radians in terms of $\pi: 200^{\circ}$

Question 9

Convert to revolutions: $3.48 \mathrm{rad}$

Question 10

Convert to revolutions: $0.492 \mathrm{rad}$

Question 11

Convert to revolutions: $7.21 \mathrm{rad}$

Question 12

Convert to revolutions: $0.165 \mathrm{rad}$

Question 13

Convert to degrees (decimal): $3.781 \mathrm{rad}$

Question 14

Convert to degrees (decimal): $1.069 \mathrm{rad}$

Question 15

Convert to degrees (decimal): $0.625 \mathrm{rad}$

Question 16

Convert to degrees (decimal): $6.30 \mathrm{rad}$

Question 17

Convert to degrees: $3 \pi / 8 \mathrm{rad}$

Question 18

Convert to degrees: $8 \pi / 7 \mathrm{rad}$

Question 19

Convert to degrees: $\pi / 3 \mathrm{rad}$

Question 20

Convert to degrees: $9 \pi / 5 \mathrm{rad}$

Question 21

Evaluate to four significant digits: $\tan 2.792$

Question 22

Evaluate to four significant digits: $\tan \pi / 9$

Question 23

Evaluate to four significant digits: $\cot 15 \pi / 7$

Question 24

Evaluate to four significant digits: $\csc 7 \pi / 5$

Question 25

Evaluate to four significant digits: $\cos^{-1} 0.1806$

Question 26

Evaluate to four significant digits: arccot 3.107

Question 27

Evaluate to four significant digits: $\sin^{2} \pi / 3+\tan \pi / 3$

Question 28

Evaluate to four significant digits: $\sin^{2} \pi / 7+\csc^{2} \pi / 7$

Question 29

Evaluate to four significant digits: $3 \cos \pi / 3+\tan^{2} \pi / 3$

Question 30

Evaluate to four significant digits: $\sec^{2} \pi / 6+3 \cot \pi / 6$

Question 31

Evaluate to four significant digits: $\cos \pi / 5+7 \sec \pi / 4$

Question 32

Evaluate to four significant digits: $\arcsin 0.871$

Question 33

Evaluate to four significant digits: $\tan^{-1} 1.36$

Question 34

Evaluate to four significant digits: $\csc^{-1} 1.36$

Question 35

Evaluate to four significant digits: $\sec^{-1} 4.50$

Question 36

Evaluate to four significant digits: $\cot^{-1} 17.1$

Question 37

Find the area of a sector with radius $6.77 \mathrm{~m}$ and central angle $70.4^{\circ}$ .

Question 38

Find the area of a segment of height 26.6 inches in a circle of radius 31.9 inches.

Question 39

Find the length of the arc $s$ subtended by a central angle $\theta=3 \pi / 4$ in a circle of radius $r=3.03 \mathrm{~m}$ .

Question 40

Find the radius $r$ of a circle where a central angle $\theta=\pi / 9$ subtends an arc length $s=21.0 \mathrm{~cm}$ .

Question 41

Find the length of the arc $s$ subtended by a central angle $\theta=108^{\circ} 29^{\prime}$ of a circle of radius $r=1.73 \mathrm{~cm}$ .

Question 42

Find the radius $r$ of a circle where a central angle $\theta=123^{\circ} 45^{\prime}$ subtends an arc length $s=9.02 \mathrm{ft}$ .

Question 43

Find the central angle $\theta$ of a circle of radius $r=4.000 \mathrm{~m}$ that subtends a length of arc $s=0.391 \mathrm{~m}$ .

Question 44

Find the central angle $\theta$ of a circle of radius $r=7.500 \mathrm{~m}$ that subtends a length of arc $s=0.755 \mathrm{~m}$ .

Question 45

Find the central angle $\theta$ of a circle of radius $r=43.5 \mathrm{~cm}$ that subtends a length of arc $s=154 \mathrm{~mm}$ .

Question 46

What arc length is subtended by a central angle of $76.5^{\circ}$ in a circle of radius $1.27 \mathrm{~m}$ ?

Question 47

What arc length is subtended by a central angle of $108.2^{\circ}$ in a circle of radius $21.2 \mathrm{~cm}$ ?

Question 48

What arc length is subtended by a central angle of $4 \pi / 9$ in a circle of radius $145 \mathrm{~m}$ ?

Question 49

The hour hand of a clock is $7.18 \mathrm{~cm}$ long. How far does the tip of the hand travel between 2 P.M. and 9 P.M.?

Question 50

Determine the latitude of a city that is 1944 miles from the equator. Assume that the earth is a sphere with radius 3960 miles.

Question 51

A $12.2 \mathrm{~m}$ long pendulum swings $4.80^{\circ}$ on each side of the vertical. Determine the length of the arc travelled by the end of the pendulum.

Question 52

An isosceles triangle is to be inscribed in a circle of radius $8.00 \mathrm{~cm}$ . Determine the angles of the triangle to degrees, minutes, and seconds if its base subtends an arc of length $11.34 \mathrm{~cm}$ .

Question 53

A circular highway curve has a radius of $100 \mathrm{~m}$ and a central angle of $15^{\circ} 45^{\prime} 30^{\prime \prime}$ (measured to the centre line of the road). What is the length of the curve?

Question 54

Calculate the area of a sector with radius $12.5 \mathrm{~cm}$ and arc length of $7.24 \mathrm{~cm}$ .

Question 55

A certain town is at a latitude of $24.9^{\circ} \mathrm{S}$ . How far is it from the North Pole? Assume that the earth is a sphere with radius $6370 \mathrm{~km}$ .

Question 56

Determine the latitude of a city that is $165 \mathrm{~km}$ from the equator. Assume that the earth is a sphere with radius $6370 \mathrm{~km}$ .

Question 57

Fill in the missing values.

Question 58

Fill in the missing values.

Question 59

Fill in the missing values.

Question 60

Calculate the angular velocity in revolutions per minute of a wheel that rotates at $83 \overline{0}$ radians per second.

Question 61

Calculate the angular velocity in degrees per second of a wheel that makes 7.5 revolutions per minute.

Question 62

A car is travelling at a rate of $110 \mathrm{~km} / \mathrm{h}$ and has tires that have a diameter of $65 \mathrm{~cm}$ . Calculate the angular velocity ( $\mathrm{rad} / \mathrm{min})$ of the wheels of the car.

Question 63

Point $A$ , which is $7.50 \mathrm{~cm}$ from the centre of the airplane propeller pictured below, is travelling at 2850 $\mathrm{cm} / \mathrm{s}$ . Determine the linear speed of point $B$ which is on the same propeller and $24.5 \mathrm{~cm}$ from the centre of the propeller.

Question 64

A fly wheel makes 650 revolutions in one minute. How many degrees does it rotate in $1.00 \mathrm{~s}$ ?

Question 65

The blades on a tidal turbine turn at 17.8 revolutions per minute. How many radians does it rotate in $45.0 \mathrm{~s}$ ?

Question 66

Olivia Oliver, an 11-year-old figure skater from Nova Scotia, performed the fastest recorded spin on ice skates on January 19, 2015, turning the equivalent of 342 revolutions per minute. How many degrees would she turn in 12.0 seconds?

Question 67

A gear is rotating at 25.0 revolutions per minute. How many seconds will it take to rotate through an angle of 1.00 rad?

Question 68

A wheel is rotating at $134.7 \mathrm{rev} / \mathrm{min}$ . Find the linear speed of a point $15.6 \mathrm{~cm}$ from the centre.

Question 69

A milling machine cutter has a diameter of $5.00 \mathrm{~mm}$ and is rotating at $42.5 \mathrm{rev} / \mathrm{min}$ . What is the linear speed at the edge of the cutter?

Question 70

A capstan on a magnetic tape drive rotates at $3250 \mathrm{rad} / \mathrm{min}$ and drives the tape at a speed of 40.0 $\mathrm{m} / \mathrm{min}$ . Find the diameter of the capstan in millimetres.

Question 71

A steel bar $16.5 \mathrm{~cm}$ in diameter is being ground in a lathe. The surface speed of the bar is $16.8 \mathrm{~m} / \mathrm{min}$ . How many revolutions will the bar make in $10 \mathrm{~s}$ ?

Question 72

A child at the edge of a spinning merry-go-round with radius 10.1 feet travels 10.0 feet in 2.0 seconds. What is the child's angular velocity in rad/sec?

Question 73

One car of a Ferris wheel of radius 25.5 feet goes from the lowest point of its circuit to the highest in 55 seconds. What is the linear speed of the car in mi./h?

Question 74

A fly wheel makes 975 revolutions in one minute. How many radians does it rotate in $6.00 \mathrm{~s}$ ?

Question 75

For peak performance, the tips of the blades on an underwater turbine should be limited to a speed of approximately $12.0 \mathrm{~m} / \mathrm{s}$ . How many revolutions per minute is this for a turbine with a diameter of $10.0 \mathrm{~m}$ ?