4.2 - Sinusoidal Derivative Application and Word Problems 2. The voltage signal from a standard North 3. Consider a simple pendulum that has a American wall socket can be described by the length of 50 cm and a maximum horizontal equation V(f) = 170sin 120n, where f is time, displacement of 8 cm. in seconds, and V(t) is the voltage, in volts, at a) Find the period of the pendulum. time t. a) Find the maximum and minimum voltage b] Determine a function that gives the horizontal position of the bob as a function levels, and the times at which they occur. of time. b) For the given signal, determine () Determine a function that gives the velocity in the period, T, in seconds of the bob as a function of time. Il the frequency, f, in hertz d) Determine a function that gives the iin the amplitude, A, in volts acceleration of the bob as a function of time. To help with #3: Recall from the lesson page that. Simple harmonic is motion that can be modelled by a sinusoidal function, and the graph of a function modelling simple harmonic motion has a constant amplitude. The horizontal position of the bob as a function of time can be described by the function where A is the amplitude of the pendulum, f is time, in seconds, I is the length of the pendulum, in metres, and y is the acceleration due to gravity. On or near the surface of Earth, g. has a constant value of 9.8 m/s. 5. A marble is placed on the end of a horizontal oscillating spring. 00000080800 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 2nft, where A is the maximum displacement from rest position, in centimetres, / is the frequency, in hertz, and * is time, in seconds. In the given situation, the spring oscillates every 1 s and has a maximum displacement of 10 cm. 1) 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. () 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.Answers: 2. al maximum voltage: 170 V at times t, in seconds, 1 = 4k + 1 240 , ke Z, k = 0 ; minimum voltage: -170 V at times I, in seconds, - +3 kez,kao 240 bjUT - 60 - $ W/ =60 Hz ID) A = 170 V 3. a) 1.412 s b) b(i) - 8 cos 1.4mr du(t) = - 11.2 sin 1.4x d) a(1) =-15.8x'cos 1.4x/ 4. a) maximum velocity: 35.2 cm/s at time f = 1.1 s b) maximum acceleration: 154.3 cm/s at time ? = 0.71 s