Applications of Sinusoidal derivatives

Applications of Sinusoidal derivatives Tuesday, April 8th

Sinusoidal derivatives • Questions from homework?

Sinusoidal derivatives • F(x) = sin3(x) What is F’(x)? 3sin2(x) 3cos2(x) 3sin2(x)cos(x) 3cos2(x)sin(x)

Sinusoidal derivatives • F’(x) = cos(2x) What is F(x)? 2sin(2x) -2sin(2x) ½sin(2x) -½sin(2x)

Sinusoidal derivatives • F(x) = cos(2x)sin(2x) What is F’(x)? 2cos2(2x) + 2sin2(2x) 2cos2(2x) – 2sin2(2x) 4sin2(2x) + 4cos2(2x) 4sin2(2x) – 4cos2(2x)

AC circuit application • This morning, I plugged my kettle into our wall, and somehow delicious tea appeared. How? • Our electricity grid uses AC (alternating current). This means that voltage flips from + to – every 0.0167s, and the current changes direction in turn. Visualizing AC

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time.

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt)

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt) Amplitude Amplitude is (max θ + min θ)/2

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt) Amplitude Amplitude = 170V

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt) Angular frequency ω = 2πf

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt) Angular frequency ω = 2π / T ω = 2πf

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = Acos(ωt) V(t) = Asin(ωt) Angular frequency ω = 380 rad/s

AC circuit application • The voltage in your wall changes from –170V to +170V and back again every 0.0167s. (You may have heard that your wall has “120V” but that’s just the root mean square or “average voltage”). • Create an equation for the voltage as a function of time. V(t) = 170sin(380t) V(t) = 170cos(380t)

AC circuit application • If V(t) = 170sin(380t), find where the voltage would be at a maximum. V(t) = 170sin(380t)

AC circuit application • If V(t) = 170sin(380t), find where the voltage would be at a maximum. V(t) = 170sin(380t) V’(t) = 64600cos(380t)

AC circuit application • If V(t) = 170sin(380t), find where the voltage would be at a maximum. V(t) = 170sin(380t) V’(t) = 64600cos(380t) 0 = 64600cos(380t) 0 = cos(380t)

AC circuit application • If V(t) = 170sin(380t), find where the voltage would be at a maximum. cos(380t) V(t) = 170sin(380t) V’(t) = 64600cos(380t) 0 = 64600cos(380t) 0 = cos(380t) time

AC circuit application • If V(t) = 170sin(380t), find where the voltage would be at a maximum. cos(380t) V(t) = 170sin(380t) V’(t) = 64600cos(380t) 0 = 64600cos(380t) 0 = cos(380t) T = 0.0167 time 0.0041s 0.012s

Applications of sinusoidal derivatives • Recall: x(t) = a function that describes position as a function of time v(t) = rate of change of position as a function of time (aka. velocity) a(t) = rate of change of velocity as a function of time (aka. acceleration) v(t) = x’(t) a(t) = v’(t) = x’’(t)

Applications of sinusoidal derivatives • New: θ(t) = a function that describes angle as a function of time ω(t) = rate of change of angle as a function of time. This is called the “angular velocity” α(t) = rate of change of angular velocity as a function of time. This is called the “angular acceleration” ω(t) = θ’(t) α(t) = ω’(t) = θ’’(t)

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle you are relative to the vertical as a function of time.

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Amplitude

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Amplitude Amplitude is (max θ + min θ)/2

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Amplitude Amplitude is 30° or ⅙π radians

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) Angular frequency θ Amplitude

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Angular frequency Alert: Angular frequency has the same letter as (and is related to) angular velocity, but these are not the same thing!

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Angular frequency Angular frequency is 2π times the number of swings that occur in one second. ω = 2πf

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Angular frequency Angular frequency: ω = 2π / T

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. In general: θ(t) = Asin(ωt) θ Angular frequency Angular frequency: ω = 2π / 4 = ½π

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • a. Construct an equation for the angle (in radians) you are relative to the vertical as a function of time. Therefore our equation is: θ(t) = ⅙π sin(½π t) θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • b. Determine the angular velocity by taking the derivative of θ with respect to time: dθ/dt. θ(t) = Asin(ωt) θ’(t) = Aωcos(ωt) θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • b. Determine the angular velocity by taking the derivative of θ with respect to time: dθ/dt. θ(t) = ⅙π sin(½π t) θ’(t) = (π2/12)cos(½π t) This is the angular velocity, ω(t) θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • b. Determine the angular acceleration. θ(t) = Asin(ωt) θ’(t) = Aωcos(ωt) θ

Applications of sinusoidal derivatives • As a calculus student, you see derivatives everywhere you go. On this occasion, you are visiting the swings at Erieau’s beach and you notice that the angle of your swing varies from +30° to -30° (where θ is the angle measured from the vertical). It takes you 4 seconds to complete one swing. • b. Determine the angular acceleration. θ(t) = Asin(ωt) θ’(t) = Aωcos(ωt) θ’’(t) = −Aω2sin(ωt) This is the angular acceleration, α(t) θ

Epic Swing design • You are designing a giant swing for our elementary school playground, and you are tasked with keeping it relatively safe for children. You decide that you should keep the maximum speed under 30m/s. • If the children will start by being pulled up by 70o (1.22 rad), what is the maximum length of the swing?

Epic Swing design • 1. Should we model the equation for angle as a cosine, sine or something in between?

Epic Swing design • Should we model the angle as a cosine, sine or something in between? • cosine – because the children will start at their maximum angle. • θ(t) = Acos(ωt)

Epic Swing design • Should we model the angle as a cosine, sine or something in between? • cosine – because the children will start at their maximum angle. • θ(t) = Acos(ωt) • θ(t) = 1.22cos(ωt) The amplitude is set by our initial angle

Epic Swing design • Should we model the angle as a cosine, sine or something in between? • cosine – because the children will start at their maximum angle. • θ(t) = Acos(ωt) • θ(t) = 1.22cos(ωt) The angular frequency will be determined by the length of the swing…

Epic Swing design • Should we model the angle as a cosine, sine or something in between? • cosine – because the children will start at their maximum angle. • θ(t) = Acos(ωt) • θ(t) = 1.22cos(ωt) The angular frequency will be determined by the length of the swing… For a pendulum length L where all the weight is at the end, the angular frequency is: ω = √(g / L)

Epic Swing design • Should we model the angle as a cosine, sine or something in between? • cosine – because the children will start at their maximum angle. • θ(t) = Acos(ωt) • θ(t) = 1.22cos(√[g/L] t) The angular frequency will be determined by the length of the swing… For a pendulum length L where all the weight is at the end, the angular frequency is: ω = √(g / L)

Epic Swing design • We can use this model to figure out the angular velocity as a function of time: • θ(t) = 1.22cos(√[g/L] t)

Epic Swing design • We can use this model to figure out the angular velocity as a function of time: • θ(t) = 1.22 cos(√[g/L] t) • θ’(t) = –1.22√[g/L] sin(√[g/L] t)

Applications of Sinusoidal derivatives