Angular vs Linear Speed
A ceiling fan has blades that are 0.4 m long (center to tip). The fan spins at 3 rad/sec.
(a) What type of speed is 3 rad/sec?
(b) Find the tip’s linear speed.
(a) Angular speed
(b) v=rω=0.4(3)=1.2v=rω=0.4(3)=1.2 m/sec
A Ferris wheel makes 1 revolution every 20 minutes.
(a) What is the period?
(b) What is the angular speed in rev/min?
(a) T = 20 min
(b) 1/20 rev/min
A horse is 10 ft from the center of a carousel.
(a) What is the amplitude of its north/south position function?
(b) What is the amplitude of its east/west position function?
(a) 10 ft
(b) 10 ft
A sinusoidal graph has a maximum value of 8 and a minimum value of 2.
(a) Find the midline.
(b) Find the amplitude.
(a) midline = (8+2)/2 = 5
(b) amplitude = (8-2)/2 = 3
A model for tide height (feet) is H(t)=2sin(0.5t)+7.
(a) Average height
(b) Max height
(c) Min height
(a) 7 ft
(b) 9 ft
(c) 5 ft
A rotating turntable has radius 0.25 m. A marker at the edge moves 0.75 m/sec.
(a) Find the angular speed (rad/sec).
(b) Is 0.75 m/sec angular or linear?
(a) ω=v/r=0.75/0.25=3ω=v/r=0.75/0.25=3 rad/sec
(b) Linear
A wheel completes 1 revolution every 24 seconds.
(a) Find angular speed in rad/sec.
(b) Write θ(t)θ(t) (t in seconds) if it starts at standard position.
(a) ω=2π/24
= π/12 rad/sec
(b) θ(t)=π/12t
A rider is 6 m from the center and completes a rotation every 12 seconds counterclockwise. Rider starts at the east-most point.
(a) Find angular speed (rad/sec).
(b) Write the rider’s east position E(t).
(a) w = 2pi/12 = pi/6 rad/sec
(b) E(t) = 6cos((pi/6)*t)
A sinusoidal function completes one full cycle over an interval of length 10.
(a) What is the period?
(b) What is the angular speed?
(a) T = 10
(b) w = 2*pi/10 = pi/5
A model for temperature is T(t)=6cos(0.4t)+50.
Explain in one sentence what 6 means and what 50 means.
6 = the temperature changes 6 above/below average
50 = the average temperature
A drone propeller has diameter 12 inches and spins at 40 rad/sec.
(a) Convert the radius to feet.
(b) Find the tip linear speed in ft/sec.
(c) How far does the tip travel in 2 seconds?
(a) Diameter 12 in → radius 6 in = 0.5 ft
(b) v=rω=0.5(40)=20v=rω=0.5(40)=20 ft/sec
(c) distance =vt=20(2)=40=vt=20(2)=40 ft
A Ferris wheel is 40 m in diameter. The center is 22 m above ground. It makes one revolution every 32 minutes.
You start tracking when you are at the lowest point.
(a) Find radius.
(b) Write height as a function of angle h(θ).
(c) Write θ(t)(t in minutes).
(d) Combine to get h(t).
a) 20 m
b) h(theta) = 22 - 20cos(theta)
c) theta(t) = (pi/16)t
d) h(t) = 22 - 20cos((pi/16)t)
A rider is 6 m from the center and completes a rotation every 12 seconds counterclockwise. Rider starts at the east-most point.
(a) Write the rider’s north position N(t).
(b) At what time does the rider first reach the north-most point?
(c) At that time, what is E(t)?
(a) N(t) = 6*sin((pi/6)*t)
(b) north-most when angle = pi/2
(pi/6)t = pi/2 -> t = 3 sec
(c) E(3) = 6cos(pi/2) = 0
A sinusoidal graph has a maximum at (2, 10) and a minimum at (6, 2).
(a) Find the midline.
(b) Find the amplitude.
(c) Find the period.
(d) Find the angular speed.
(a) midline = (10 + 2)/2 = 6
(b) amplitude = (10 - 2)/2 = 4
(c) half-period = 6 - 2 = 4 → T = 8
(d) w = 2*pi/8 = pi/4
Beaver population model: P(t)=180sin(0.6t)+520, t in years.
(a) Max population
(b) Min population
(c) Time from one max to the next max (period, in years)
(a) 520+180 = 700
(b) 520-180 = 340
(c) T = 2pi/0.6 = 10pi/3 years
A lab centrifuge has radius 0.15 m. It spins at 1200 rev/min.
(a) Convert angular speed to rad/min.
(b) Convert to rad/sec.
(c) Find the linear speed (m/sec).
(a) 1200(2π)=2400π1200(2π)=2400π rad/min
(b) 2400π/60=40π2400π/60=40π rad/sec
(c) v=rω=0.15(40π)=6πv=rω=0.15(40π)=6π m/sec
A Ferris wheel has radius 18 ft, center 25 ft above ground, and period 30 sec.
You start tracking at the height of the center and you are moving upward.
(a) Find angular speed in rad/sec.
(b) Pick sine or cosine (no phase shift) that matches “center height & moving up.”
(c) Write h(t).
(d) Find your height at t = 7.5 sec.
(a) w = 2pi/30 = pi/15 rad/sec
(b) sine (starts at midline and goes up)
(c) h(t) = 18sin((pi/15)*t) + 25
(d) angle = (pi/15)7.5 = pi/2
sin(pi/2) = 1
h = 181 + 25 = 43 ft
A point on a carousel is 12 ft from the center. It completes a rotation every 15 sec CCW.
You start tracking at the south-most point.
(a) Find w.
(b) Write N(t) (north position).
(c) Find the first time N(t) = 6.
(a) w = 2pi/15 rad/sec
(b) N(t) = -12cos((2pi/15)t)
(c) -12cos((2pi/15)t) = 6
cos((2pi/15)t) = -1/2
first time: (2pi/15)t = 2pi/3
t = 5 sec
A sinusoidal function has:
midline = 4
amplitude = 3
period = 5
it starts at a minimum when x = 0
(a) Which function fits without phase shift: sine, cosine, negative sine, or negative cosine?
(b) Write the function.
(a) negative cosine
(b) w = 2pi/5
y = -3cos((2*pi/5)*x) + 4
A daylight model (hours) is D(m)=3.4sin((pi/6)m)+12.2, where m is months after January.
(a) Period in months
(b) Which months are predicted to be brightest (max daylight)?
(c) Is a period of 14 months reasonable? Explain.
(a) T = 2pi/(pi/6) = 12 months
(b) max when (pi/6)m = pi/2 + 2pi*k -> m = 3 + 12k (about April each year)
(c) no, it repeats yearly so 12 months makes sense
A circular saw has radius 0.6 ft. The tip is moving 18 ft/sec.
(a) Find angular speed in rad/sec.
(b) Convert to rev/min.
(c) Find the period (seconds per revolution).
(a) ω=v/r=18/0.6=30ω=v/r=18/0.6=30 rad/sec
(b) rev/sec =30/(2π)=15/π=30/(2π)=15/π. rev/min =(15/π)60=900/π=(15/π)60=900/π rev/min
(c) T=1/(15/π)=π/15T=1/(15/π)=π/15 sec
A wheel has diameter 52 m. The platform at the bottom is 2 m above ground. It rotates once every 26 minutes.
A rider starts at the bottom.
(a) Find radius and center height.
(b) Write h(theta).
(c) Write h(t) (t in minutes).
(d) Solve for the first time the rider reaches 30 m.
(a) r = 26 m
center height = 2 + 26 = 28 m
(b) h(theta) = 28 - 26cos(theta)
(c) w = 2pi/26 = pi/13 rad/min
h(t) = 28 - 26*cos((pi/13)t)
(d) 30 = 28 - 26cos((pi/13)t)
2 = -26cos((pi/13)*t)
cos((pi/13)*t) = -1/13
first time: (pi/13)*t = arccos(-1/13)
t = (13/pi)*arccos(-1/13) minutes
A rider is 9 m from the center, rotates CCW with period 18 sec. Tracking starts when the rider is 3 m east of center and moving toward the north.
(a) Find w.
(b) Write E(t) and N(t) (you will need a phase shift).
(c) Find the first time the rider is at the west-most point.
(a) w = 2pi/18 = pi/9 rad/sec
(b) E(t) = 9cos((pi/9)t + arccos(1/3))
N(t) = 9sin((pi/9)*t + arccos(1/3))
(c) west-most means E(t) = -9
cos((pi/9)*t + arccos(1/3)) = -1
(pi/9)t + arccos(1/3) = pi
t = (9/pi)(pi - arccos(1/3)) sec
A sinusoidal function has sequential maximum and minimum points at (1, 12) and (5, 4).
(a) Find the midline.
(b) Find the amplitude.
(c) Find the period.
(d) Find the angular speed.
(e) Write a sine model for the function.
(a) midline = (12 + 4)/2 = 8
(b) amplitude = (12 - 4)/2 = 4
(c) half-period = 5 - 1 = 4 → T = 8
(d) w = 2pi/8 = pi/4
(e) y = 4sin((pi/4)*(x - 1) + pi/2) + 8
A sunspot prediction model is S(m)=-60cos((pi/48)(m-10))+70, with m months after Jan 2020.
(a) Predicted max and min sunspots
(b) Period in months and in years
(c) Predicted sunspots in September 2021 (that is 20 months after Jan 2020)
(d) If real data in that month is way higher than predicted, which model change would help most: midline up/down, amplitude up/down, or period longer/shorter?
(a) max = 70+60 = 130, min = 70-60 = 10
(b) T = 2*pi/(pi/48) = 96 months = 8 years
(c) S(20) = -60cos((pi/48)(20-10)) + 70
= -60cos((pi/48)10) + 70
= -60cos(5pi/24) + 70