4.4 Applications of Sinusoidal Functions and Their Derivatives
Chapter
Chapter 4
Section
4.4
Solutions 15 Videos

An AC-DC coupled circuit produces a current described by the function I(t) = 60cost + 25, where t is time, in seconds, and I is the current, in amperes, at time t.

a) Find the maximum and minimum currents and the times at which they occur.

b) For the given current, determine

• i) the period, T, in seconds
• ii) the frequency, f, in hertz
• iii) the amplitude, A, in amperes
Q1

The voltage signal from a standard North American wall socket can be described by the equation V(t) = 170\sin 120\pi t, where t is time, in seconds, and V is the voltage, in volts, at time t.

a) Find the maximum and minimum voltage levels and the times at which they occur.

b) For the given signal, determine

• i) the period, $T$, in seconds
• ii) the frequency, f, in hertz
• iii) the amplitude, A, in volts
Q2

Consider a simple pendulum that has a length of 50 cm and a maximum horizontal displacement of 8 cm.

a) Find the period of the pendulum.

b) Determine a function that gives the horizontal position of the bob as a function of time.

c) Determine a function that gives the velocity of the bob as a function of time.

d) Determine a function that gives the acceleration of the bob as a function of time.

Q3

Consider a simple pendulum that has a length of 50 cm and a maximum horizontal displacement of 8 cm.

a) Find the maximum velocity of the bob and the time at which it first occurs.

b) Find the maximum acceleration of the bob and the time at which it first occurs.

c) Determine the times at which

• i) the displacement equals zero
• ii) the velocity equals zero
• iii) the acceleration equals zero

d) Describe how the answers in part c) are related in terms of when they occur. Explain Why these results make sense.

Q4

A marble is placed on the end of a horizontal oscillating spring.

If you ignore the effect of friction and treat this situation as an instance of simple harmonic motion, the horizontal position of the marble as a function of time is given by the function b(t) = A cos 2\pi t, where A is the maximum displacement from rest position, in centimetres, f is the frequency, in hertz, and t is time, in seconds. In the given situation, the spring oscillates every 1 s and has a maximum displacement of 10 cm.

a) What is the frequency of the oscillating spring?

b) Write the simplified equation that expresses the position of the marble as a function of time.

c) Determine a function that expresses the velocity of the marble as a function of time.

d) Determine a function that expresses the acceleration of the marble as a function of time.

Q5

a) Sketch a graph of each of the following relations over the interval from O to 4 s. Align your graphs vertically.

• i) displacement versus time
• ii) velocity versus time
• iii) acceleration versus time

b) Describe any similarities and differences between the graphs. Find the maximum and minimum values for displacement. When do these values occur? Refer to the other graphs and explain why these results make sense.

Q6

A piston in an engine oscillates up and down from a rest position as shown.

The motion of this piston can be approximated by the function b(t) = 0.05\cos(13t), where t is time, in seconds, and la is the displacement of the piston head from rest position, in metres, at time t.

a) Determine an equation for the velocity of the piston head as a function of time.

b) Find the maximum and minimum velocities and the times at which they occur.

Q7

A high—power distribution line delivers an AC—DC coupled voltage signal whose

• AC component has an amplitude, A, of 380 kV
• DC component has a constant voltage, V, of 120 kV
• frequency, f, is 60 Hz

a) Add the Ac component, V_{AC}, and DC component, V_{DC}, to determine an equation that relates voltage, V, in kilovolts, to time, t, in seconds.

Use the equation V_{AC}(t) = A \sin 2\pi ft to determine the AC component.

b) Determine the maximum and minimum voltages and the times at which they occur.

Q8

A differential equation is an equation involving a function and one or more of its derivatives. Determine whether the function y = \pi \sin \theta + 2\pi \cos \theta is a solution to the equation \displaystyle \frac{d^2y}{d\theta^2} + y = 0 .

Q9

a) Determine a function that satisfies the differential equation \displaystyle \frac{d^2y}{dx^2} = - 4y .

b) Explain how you found your solution.

Q10

An oceanographer during a storm and modelled the vertical displacement of a wave, in metres, using the equation h(t) = 0.6\cos 2t + 0.8\sin t, where t is the time in seconds.

a) Determine the vertical displacement of the wave when the velocity is 0.8 m/s.

b) Determine the maximum velocity of the wave and when it first occurs.

c) When does the wave first change from a “hill” to a “trough”? Explain.

Q12

Potential energy is energy that is stored, for example, the energy stored in a compressed or extended spring. The amount of potential energy stored in a spring is given by the equation \displaystyle U = \frac{1}{2}k x^2 , where

• U is the potential energy, in joules
• k is the spring constant, in newtons per metre
• x is the displacement of the spring from rest position, in metres

Use the displacement equation from question 5 to find the potential energy of an oscillating spring as a function of time.

Q13

Kinetic energy is the energy of motion. The kinetic energy of a spring is given by the equation \displaystyle K = \frac{kv^2 T^2}{8\pi^2} , where K is the kinetic energy, in joules; k is the spring constant, in newtons per metre; v is the velocity as a function of time, in metres per second; and T is the period, in seconds.

Use the velocity equation from question 5 to express the kinetic energy of an oscillating spring as a function of time.

Q14

An oscillating spring has a spring constant of 100 N/s, an amplitude of 0.02 m, and a period of 0.5s.

a) Graph the function relating potential energy to time in this situation. Find the maxima, minima, and zeros of the potential energy function, and the times at which they occur.

b) Repeat part a) for the function relating kinetic energy to time.

c) Explain how the answers to parts a) and b) are related,

Q15

For any constants A and B, the local maximum value of A \sin x + B \cos x is

A. \displaystyle \frac{1}{2}|A + B| 

B. \displaystyle A + B| 

C. \displaystyle \frac{1}{2}(|A| + |B)| 

D. \displaystyle |A| + |B) 

E. \displaystyle \sqrt{A^2 + B^2} `