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A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight move- vertically from its equilibrium position, and this motion is modeled by y = - sin(4t) + cos(4t) 4 where y is the displacement (in feet) from equilibrium of the weight and t is the time (in seconds). (a) Use the identity a sin(B0) + b cos(Be) = v a² + b²2 sin(B0 + C) where C = arctan(b/a), a > 0, to write the model in the form y = Va2 + b2 sin(Bt + C), (Round C to four decimal places.) y = (b) Find the amplitude of the oscillations of the weight. ft (c) Find the frequency of the oscillations of the weight. cycle per second 6042021

A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight move- vertically from its equilibrium position, and this motion is modeled by y = - sin(4t) + cos(4t) 4 where y is the displacement (in feet) from equilibrium of the weight and t is the time (in seconds). (a) Use the identity a sin(B0) + b cos(Be) = v a² + b²2 sin(B0 + C) where C = arctan(b/a), a > 0, to write the model in the form y = Va2 + b2 sin(Bt + C), (Round C to four decimal places.) y = (b) Find the amplitude of the oscillations of the weight. ft (c) Find the frequency of the oscillations of the weight. cycle per second 6042021

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 98E
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--- ### Oscillatory Motion of a Weight on a Spring A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by the equation: \[ y = \frac{1}{3} \sin(4t) + \frac{1}{4} \cos(4t) \] where \( y \) is the displacement (in feet) from the equilibrium of the weight and \( t \) is the time (in seconds). #### (a) Use the identity \[ a \sin(B\theta) + b \cos(B\theta) = \sqrt{a^2 + b^2} \sin(B\theta + C) \] where \( C = \arctan(b/a) \), \( a > 0 \), to write the model in the form: \[ y = \sqrt{a^2 + b^2} \sin(Bt + C) \] (Round \( C \) to four decimal places.) \[ y = \] #### (b) Find the amplitude of the oscillations of the weight. \[ \text{Amplitude} = \quad \text{ft} \] #### (c) Find the frequency of the oscillations of the weight. \[ \text{Frequency} = \quad \text{cycle per second} \] --- ### Explanation of the Equations: - **Given Equation:** The given equation describes the displacement \( y \) of the weight at any time \( t \). - **Identity Used:** To simplify this kind of trigonometric expression, we use the identity combining sine and cosine into a single sine function with a phase shift. - **Amplitude and Frequency:** Once the equation is in the simplified form, the amplitude of the oscillations will be \( \sqrt{a^2 + b^2} \), and the frequency relates to the coefficient of \( t \) in the sine function. This theoretical approach is fundamental in understanding oscillatory motion, particularly in systems like springs, pendulums, and even electrical circuits exhibiting harmonic motion.
Transcribed Image Text:--- ### Oscillatory Motion of a Weight on a Spring A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by the equation: \[ y = \frac{1}{3} \sin(4t) + \frac{1}{4} \cos(4t) \] where \( y \) is the displacement (in feet) from the equilibrium of the weight and \( t \) is the time (in seconds). #### (a) Use the identity \[ a \sin(B\theta) + b \cos(B\theta) = \sqrt{a^2 + b^2} \sin(B\theta + C) \] where \( C = \arctan(b/a) \), \( a > 0 \), to write the model in the form: \[ y = \sqrt{a^2 + b^2} \sin(Bt + C) \] (Round \( C \) to four decimal places.) \[ y = \] #### (b) Find the amplitude of the oscillations of the weight. \[ \text{Amplitude} = \quad \text{ft} \] #### (c) Find the frequency of the oscillations of the weight. \[ \text{Frequency} = \quad \text{cycle per second} \] --- ### Explanation of the Equations: - **Given Equation:** The given equation describes the displacement \( y \) of the weight at any time \( t \). - **Identity Used:** To simplify this kind of trigonometric expression, we use the identity combining sine and cosine into a single sine function with a phase shift. - **Amplitude and Frequency:** Once the equation is in the simplified form, the amplitude of the oscillations will be \( \sqrt{a^2 + b^2} \), and the frequency relates to the coefficient of \( t \) in the sine function. This theoretical approach is fundamental in understanding oscillatory motion, particularly in systems like springs, pendulums, and even electrical circuits exhibiting harmonic motion.
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