?Graphing by Calculator For the combinations of sine and cosine functions in 9-13, do

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot x + x =0, x(0)=1. Dot x(0)=0

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot + x =0, x(0)=1, dot x(0)=1

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+9x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot + 9x =0, x(0)=1, dot x(0)=1

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+4 x=0, \quad x(0)=1, \quad \dot{x}(0)=-2\) ________________ Equation Transcription: Text Transcription: Double dot + 4x =0, x(0)=1, dot x(0)=-2

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+16 x=0, \quad x(0)=-1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot + 16x =0, x(0)=-1, dot x(0)=0

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+16 x=0, \quad x(0)=0, \quad \dot{x}(0)=4\) ________________ Equation Transcription: Text Transcription: Double dot + 16x =0, x(0)=0, dot x(0)=4

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(\ddot{x}+16 \pi^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: Double dot + 16pi^2 =0, x(0)=0, dot x(0)=pi

The Undamped Oscillator For Problems 1-8, Find the simple harmonic motion described by the initial-value problem. See also Problems 23-30 and 32-39. \(4 \ddot{x}+\pi^{2} x=0, \quad x(0)=1, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: 4 Double dot + pi^2 x=0, x(0)=1, dot x(0)=pi

Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulting oscillation (c) Write each function in the form A cos(\(\omega_{0} t-\delta\)). \(x(t)=\cos t+\sin t\) ________________ Equation Transcription: / t - Text Transcription: delta/omega_0 Omega_0 t- delta x(t) =cos t+ sin t

Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulting oscillation (c) Write each function in the form A cos(\(\omega_{0} t-\delta\)). \(x(t)=2 \cos t+\sin t\) ________________ Equation Transcription: / t - Text Transcription: delta/omega_0 Omega_0 t- delta x(t) =2cos t+ sin t

Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulting oscillation (c) Write each function in the form A cos(\(\omega_{0} t-\delta\)). \(x(t)=5 \cos 3 t+\sin 3 t\) ________________ Equation Transcription: / t - Text Transcription: delta/omega_0 Omega_0 t- delta x(t) = 5cos 3t + sin 3t

Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulting oscillation (c) Write each function in the form A cos(\(\omega_{0} t-\delta\)). \(x(t)=\cos 3 t+5 \sin 3 t\) ________________ Equation Transcription: / t - Text Transcription: delta/omega_0 Omega_0 t- delta x(t)=cos 3t+5sin 3t

Graphing by Calculator For the combinations of sine and cosine functions in Problems 9-13, do the following. (a) Use a graphing calculator or computer to sketch the graph of each function. (b) From your graphs, estimate the amplitude. period, and phase shift \(\delta / \omega_{0}\) of the resulting oscillation (c) Write each function in the form A cos(\(\omega_{0} t-\delta\)). \(x(t)=-\cos 5 t+2 \sin 5 t\) ________________ Equation Transcription: / t - Text Transcription: delta/omega_0 Omega_0 t- delta x(t) = -cos 5t +2sin 5t

Alternate Forms for Sinusoidal Oscillations To derive the conversion formulas (7) and (8). use the identity \(\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\) from trigonometry to show that the family of sinusoidal oscillations A \(\cos \left(\omega_{0} t-\delta\right)\) can be written in the form \(c_{1} \cos \omega_{0} t+c_{2} \sin \omega_{0} t\) where \(C_{1}=A \cos \delta\) and \(c_{2}=A \sin \delta\) ________________ Equation Transcription: cos( - ) = cos cos + sin sin cos( - ) c1 cos + c2 sin c1 = A cos c2 = A sin Text Transcription: cos( alpha - beta) = cos alpha cos beta + sin alpha sin beta cos(Omega_0t - delta) c_1 cos Omega_0t - delta + c_2 sin Omega_0t - delta c_1 = A cos delta c_2 = A sin delta

Single-Wave Forms of Simple Harmonic Motion Rewrite Problems 15-18 in the form A cos( \(\omega_{0} t-\delta\)) using the conversion equations (7). \(\cos t+\sin t\) ________________ Equation Transcription: - Text Transcription: Omega_0t - delta cos t+sin t

Single-Wave Forms of Simple Harmonic Motion Rewrite Problems 15-18 in the form A cos( \(\omega_{0} t-\delta\)) using the conversion equations (7). \(\cos t-\sin t\) ________________ Equation Transcription: - Text Transcription: Omega_0t - delta cos t-sin t

Single-Wave Forms of Simple Harmonic Motion Rewrite Problems 15-18 in the form A cos( \(\omega_{0} t-\delta\)) using the conversion equations (7). \(-\cos t+\sin t\) ________________ Equation Transcription: - Text Transcription: Omega_0t - delta -cos t+sin t

Single-Wave Forms of Simple Harmonic Motion Rewrite Problems 15-18 in the form A cos( \(\omega_{0} t-\delta\)) using the conversion equations (7). \(-\cos t-\sin t\) ________________ Equation Transcription: - Text Transcription: Omega_0t - delta -cos t-sin t

Component Form of Simple Harmonic Motion Rewrite Problems 19-22 in the form \(\mathrm{c}_{1} \cos \omega_{0} t+\mathrm{c}_{2} \sin \omega_{0} t\) using the conversion equations (8). \(2 \cos (2 t-\pi)\) ________________ Equation Transcription: c1 cos + c2 sin 2 cos(2t -) Text Transcription: c_1 cos omega_0 t + c_2 sin omega_0 t 2 cos(2t-pi)

Component Form of Simple Harmonic Motion Rewrite Problems 19-22 in the form \(\mathrm{c}_{1} \cos \omega_{0} t+\mathrm{c}_{2} \sin \omega_{0} t\) using the conversion equations (8). \(\cos \left(t+\frac{\pi}{3}\right)\) ________________ Equation Transcription: c1 cos + c2 sin Text Transcription: c_1 cos omega_0 t + c_2 sin omega_0 t cos(t+pi/3)

Component Form of Simple Harmonic Motion Rewrite Problems 19-22 in the form \(\mathrm{c}_{1} \cos \omega_{0} t+\mathrm{c}_{2} \sin \omega_{0} t\) using the conversion equations (8). \(3\cos \left(t-\frac{\pi}{4}\right)\) ________________ Equation Transcription: c1 cos + c2 sin Text Transcription: c_1 cos omega_0 t + c_2 sin omega_0 t 3 cos(t-pi/4)

Component Form of Simple Harmonic Motion Rewrite Problems 19-22 in the form \(\mathrm{c}_{1} \cos \omega_{0} t+\mathrm{c}_{2} \sin \omega_{0} t\) using the conversion equations (8). \(\cos \left(3t-\frac{\pi}{6}\right)\) ________________ Equation Transcription: c1 cos + c2 sin Text Transcription: c_1 cos omega_0 t + c_2 sin omega_0 t cos(t-pi/6)

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot x + x = 0, x(0)=1, dot x(0)=0

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot x + x = 0, x(0)=1, dot x(0)=1

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+9x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot x + 9x = 0, x(0)=1, dot x(0)=1

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+4x=0, \quad x(0)=1, \quad \dot{x}(0)=-2\) ________________ Equation Transcription: Text Transcription: Double dot x + 4x = 0, x(0)=1, dot x(0)=-2

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+16x=0, \quad x(0)=-1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot x + 16x = 0, x(0)=-1, dot x(0)=0

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+16x=0, \quad x(0)=0, \quad \dot{x}(0)=4\) ________________ Equation Transcription: Text Transcription: Double dot x + 16x = 0, x(0)=0, dot x(0)=4

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(\ddot{x}+16 \pi^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: Double dot x + 16 pi^2 x = 0, x(0)=0, dot x(0)=pi

Interpreting Oscillator Solutions For Problems 23-30. determine the amplitude, phase angle, and period of the motion (These are the equations of Problems 1-8 and 32-39.) \(4 \ddot{x}+\pi^{2} x=0, \quad x(0)=1, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: 4 Double dot x + pi^2 x = 0, x(0)=1, dot x(0)=pi

Relating Graphs For the oscillator DE \(\ddot{x}+0.25 x=0\) Fig. 4.1.4 shown previously linked solution graphs and phase portrait. Parts (a), (b), (c), and (d) relate to that figure. (a) Mark on the phase portrait the starting points (where t = 0) for the trajectories shown. (b) Write explicit solutions for \(x(t) \text { and } \dot{x}(t)\). What is the value of ? (c) Label the t axis in the solutions graphs of Fig.4. l .4with the appropriate values for HINT: Consider where the solution graphs cross the axis. (d) Given A as the amplitude of the solution with the largest oscillation, state the amplitudes of the other solutions shown. ________________ Equation Transcription: x(t) and ?(t) Text Transcription: Double x + 0.25x=0 x(t) and dot x(t) omega_0

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot x + x = 0 x(0)=1, dot x(0)=0

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot x + x = 0 x(0)=1, dot x(0)=1

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+9x=0, \quad x(0)=1, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double dot x + 9x = 0 x(0)=1, dot x(0)=1

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+4x=0, \quad x(0)=1, \quad \dot{x}(0)=-2\) ________________ Equation Transcription: Text Transcription: Double dot x + 4x = 0 x(0)=1, dot x(0)=-2

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+16x=0, \quad x(0)=-1, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double dot x + 16x = 0 x(0)=-1, dot x(0)=0

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+16x=0, \quad x(0)=0, \quad \dot{x}(0)=4\) ________________ Equation Transcription: Text Transcription: Double dot x + 16x = 0 x(0)=0, dot x(0)=4

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(\ddot{x}+16 \pi^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: Double dot x + 16 pi^2 x = 0 x(0)=0, dot x(0)=pi

Phase Portraits For Problems 32-39. find ?(t) and then sketch the trajectory for the IVP in the x? phase plane, with arrows showing the direction of motion. (These are the equations of Problems 1-8 and 23-30. ) Explain how and why your phase portraits differ from each other and from Fig. 4.1.4. \(4 \ddot{x}+\pi^{2} x=0, \quad x(0)=1, \quad \dot{x}(0)=\pi\) ________________ Equation Transcription: Text Transcription: Double dot x + pi^2 x = 0 x(0)=1, dot x(0)=pi

Matching Problems Match the IVPs in Problems 40-43 to the graphs in Fig. 4.1.7. \(\ddot{x}+4 x=0, \quad x(0)=0, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double + 4x=0 x(0)=0, dot x(0)=1

Matching Problems Match the IVPs in Problems 40-43 to the graphs in Fig. 4.1.7. \(\ddot{x}+4 x=0, \quad x(0)=0, \quad \dot{x}(0)=0\) ________________ Equation Transcription: Text Transcription: Double + 4x=0 x(0)=1, dot x(0)=0

Matching Problems Match the IVPs in Problems 40-43 to the graphs in Fig. 4.1.7. \(\ddot{x}+\pi^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: Double + pi^2 x=0 x(0)=0, dot x(0)=1

Matching Problems Match the IVPs in Problems 40-43 to the graphs in Fig. 4.1.7. \(4 \ddot{x}+\pi^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=1\) ________________ Equation Transcription: Text Transcription: 4 Double + pi^2 x=0 x(0)=0, dot x(0)=1

Changing Frequencies Consider the undamped harmonic oscillator defined by \(\ddot{x}+\omega_{0}^{2} x=0\) with initial conditions \(x(0)=4 \text { and } \dot{x}(0)=0\) (a) For \(\omega_{0}=0.5,1 \text {, and } 2\), the corresponding trajectories are plotted in Fig. 4.1.8(a) on the same tx-plane. In Fig. 4.1.803) the corresponding trajectories are plotted on the x? phase plane. Make a trace of both graphs and label each curve with the appropriate value of \(\omega_{0}\). (b) If \(\omega_{0}\) is increased. we can see that the frequency of the oscillations in Fig. 4. l .8(a) increases (as expected. since \(\omega_{0}\) is the natural (circular) frequency). Describe what happens in the phase plane (F ig. 4.1.8(b)) if \(\omega_{0}\) is increased. How is ? affected if \(\omega_{0}\) is increased? ________________ Equation Transcription: x(0)=4 and ?(0)=0 = 0.5, 1, and 2 Text Transcription: Double dot x + omega_0^2 x=0 x(0)=4 and dot x(0)=0 omega=0.5,1, and 2 omega_0

Detective Work Suppose that you have received the following two graphs without their equations. Show how you can infer the equations from graphical information. (a) The graph in Fig. 4.1.9(a) represents \(A \cos (t-\delta)\). Determine A and \(\delta\) from the graph and write the equation of the curve in the fonn of equation (5). (b) The graph in Fig. 4.1.9(b)represents \(c_{1} \cos t+c_{2} \sin t\). Determine \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) from the graph and write the equation of the curve in the form of (4). HINT: Find A and \(\delta\) first. ________________ Equation Transcription: A cos(t -) c1 cos t+c2 sin t c1 c2 Text Transcription: A cos(t-delta) Delta C_1 cos t + c_2 sin t C_1 c_2

Pulling a Weight An object of mass \(2 kg\), resting on a frictionless table. is attached to the wall by a spring as in Fig 4.1.1. A force of 8 nt is applied to the mass. Stretching the spring and moving the mass \(0.5 m\) from its equilibrium position. The object is then released. (a) Find the resulting motion of the object as a function of time (b) Determine the amplitude. period. and frequency of the motion. (c) At what time does the mass first pass through the equilibrium position? What is its velocity at that time? ________________ Equation Transcription: Text Transcription: 2 kg 0.5 m

Finding the Differential Equation A mass of \(500 gm\) is suspended from the ceiling by a frictionless spring. The mass stretches the spring \(50 cm\) in coming to its equilibrium position. where the mass acting down is balanced exactly by the restoring force acting up. The object is then pulled down an additional \(10 cm\) and released. (a) Formulate the initial-value problem that describes the object's motion. setting x equal to the downward displacement from equilibrium. (b) Solve for the motion of the object. (c) Find the amplitude. phase angle. frequency. and period of the motion. ________________ Equation Transcription: Text Transcription: 500 gm 50 cm 10 cm

initial-Value Problems A \(16-lb\) object is attached to the ceiling by a frictionless spring and stretches the spring \(6 in\). before coming to its equilibrium position. Formulate the initial value problem describing the motion of the object under each of the following sets of conditions. Set x equal to the downward displacement from equilibrium. (a) The object is pulled down \(4 in\). below its equilibrium position and released with an upward velocity of \(4 ft/sec\). (b) The object is pushed up \(2 in\). and released with a downward velocity of \(1 ft/sec\). ________________ Equation Transcription: Text Transcription: 16-lb 6 in 4 in 4 ft/sec 2 in 1 ft/sec

One More Weight A \(12-lb\) object attached to the ceiling by a frictionless spring stretches the spring \(6 in\). as it comes to its equilibrium. Find and solve the equation of motion if the object is initially pushed up \(4 in\). from its equilibrium and given an upward velocity of \(2 ft/sec\). ________________ Equation Transcription: Text Transcription: 12-lb 6 in 4 in 2 ft/sec

Comparing Harmonic Motions An object on a table attached to spring and wall as in Fig. 4.1.1 is pulled to the right. stretching the spring. and released The same object is then pulled twice as far and released What is the relationship between the two simple harmonic motions? Will the period of the second be twice that of the first? What about the amplitudes and frequencies?

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}+x+x^{3}=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x + x + x^3=0

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}+x-x^{3}=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x + x - x^3=0

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}-x=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x - x = 0

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}+\frac{1}{t} \dot{x}+x=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x + 1/t + x = 0

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}+\left(x^{2}-1\right) \dot{x}+x=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x + (x^2-1) dot x + x =0

Testing Your Intuition Knowing (from Example I) what you now do about the damped harmonic oscillator equation \(m \ddot{x}+b \dot{x}+k x=0\) and the meaning of the parameters m. b. and k, consider Problems 51-56. How would you expect the solution of each equation to behave? Can you imagine a physical system being modeled by the equation? What would you expect for its long-term behavior? \(\ddot{x}+t x=0\) ________________ Equation Transcription: Text Transcription: m double dot x + b dot x + kx =0 Double dot x + tx=0

LR-Circuit Consider the series LR-circuit shown in Fig. 4.1.10, in which a constant input voltage \(\mathrm{V}_{0}\) has been supplied until \(t = 0\). when it is shut off. (a) Before carrying out the mathematical analysis, describe (b) For \(t > 0\), use Kirchoff's voltage law to determine the sum of the voltage drops around the circuit. Set it equal to zero to obtain a first-order differential equation involving R, I, L, and I. What are the initial conditions (c) Solve the DE in (b) for current I. Does your answer agree with (a)? Explain. (d) Use the values \(R = 40 ohms\), \(L = 5 henries\), and \(V_{0}=10 \text { volts }\) to obtain an explicit solution. ________________ Equation Transcription: V0 V0 Text Transcription: V_0 t = 0 t > 0 R = 40 ohms L = 5 henries V_0 = 10 volts

LC-Circuit Consider the series LC-circuit shown in Fig. 4.1.11 , in which, at \(t = 0\), the current is \(5 amps\) and there is no charge on the capacitor. Voltage \(V_{0}\) is turned off at \(t = 0\). (a) Before carrying out the mathematical analysis, describe what you think will happen to the charge on the capacitor. (b) For \(t > 0\), use Kirchoff's voltage law to determine the sum of the voltage drops around the circuit. Set it equal to zero to obtain a second-order differential equation involving L, Q, Q, and C. What are the initial conditions? (c) Solve the IVP in (b) for the charge Q on the capacitor. Does your result agree with pan (a)? (d) Obtain an explicit solution for \(L = 10 henries\) and \(C=10^{-3} \text { farads }\). ________________ Equation Transcription: V0 C = 10-3 farads Text Transcription: t = 0 5 amps V_0 t > 0 L = 10 henries C = 10^-3 farads

A Pendulum Experiment A pendulum of length L is suspended from the ceiling so it can swing freely; \(\theta\) denotes the angular displacement, in radians, from the vertical, as shown in Fig. 4.1.1 2. The motion is described by the pendulum equation, \(\theta+\frac{8}{L} \sin \theta=0\). Determine the period for small oscillations by using the approximation sin \(\theta \approx \theta\) (the linear pendulum). What is the relationship between the period of the pendulum and g, the acceleration due to gravity? If the sun is 400,000 times more massive than the earth, how much faster would the pendulum oscillate on the sun (provided it did not melt) than on the earth? ________________ Equation Transcription: Text Transcription: Theta theta+8/L sin theta=0 Theta approx theta

Changing into Systems For Problems 60-64, consider the second-order nonhomogeneous DEs. Write them as a system of first-order DEs as in (18). \(4 \ddot{x}-2 \dot{x}+3 x=17-\cos t\) ________________ Equation Transcription: Text Transcription: 4 double dot x - 2 dot x + 3x =17 - cos t

Changing into Systems For Problems 60-64, consider the second-order nonhomogeneous DEs. Write them as a system of first-order DEs as in (18). \(L \ddot{q}+R \dot{q}+\frac{1}{c} q=V(t)\) ________________ Equation Transcription: Text Transcription: L double dot q + R dot q + 1/c q =V(t)

Changing into Systems For Problems 60-64, consider the second-order nonhomogeneous DEs. Write them as a system of first-order DEs as in (18). \(5 \ddot{q}+15 \dot{q}+\frac{1}{10} q=5 \cos 3 t\) ________________ Equation Transcription: Text Transcription: 5 double dot q + 15 dot q + 1/10 q =5 cos 3t

Changing into Systems For Problems 60-64, consider the second-order nonhomogeneous DEs. Write them as a system of first-order DEs as in (18). \(t^{2} \ddot{x}+4 t \dot{x}+x=t \sin 2 t\) ________________ Equation Transcription: Text Transcription: t^2 double dot x + 4t dot x + x = t sin 2t

Changing into Systems For Problems 60-64, consider the second-order nonhomogeneous DEs. Write them as a system of first-order DEs as in (18). \(4 \ddot{x}+16 x=4 \sin t\) ________________ Equation Transcription: Text Transcription: 4 double dot x + 16x = 4 sin t

Circular Motion A particle moves around the circle \(x^{2}+y^{2}=r^{2}\) with a constant angular velocity of \(\omega_{0}\) radians per unit time. Show that the projection of the particle on the x axis satisfies the equation \(\ddot{x}+\omega_{0} x=0\). ________________ Equation Transcription: x2 + y2 = r2 Text Transcription: x^x+y^2=r^2 Omega_0 Double dot x + omega_0 x = 0

Another Harmonic Motion The mass-sprint-pulley system shown in Fig. 4. 1 . 1 3 satisfies the differential equation \(\ddot{x}+\left(\begin{array}{c} k R^{2} \\ m R^{2}+1 \end{array}\right) x=0, \) where x is the displacement from equilibrium of the object of mass m. In this equation, R and I are, respectively, the radius and moment of inertia of the pulley, and k is the spring constant. Determine the frequency of the motion. ________________ Equation Transcription: Text Transcription: Double dot x + (kR^2/ mR^2 + 1)x =0

Motion of a Buoy A cylindrical buoy with diameter \(18 in\). floats in water with its axis vertical, as shown in Fig. 4.1.14. When depressed slightly and released, its period of vibration is found to be 2.7 sec. Find the weight of the cylinder. HINT: Archimedes' Principle says that an object submerged in water is buoyed up by a force equal to the weight of the water displaced, where weight is the product of volume and density. The density of water is 62.5 lb/ft3. ________________ Equation Transcription: Text Transcription: 18 in

Los Angeles to Tokyo It can be shown that the force on an object inside a spherical homogeneous mass is directed towards the center of the sphere with a magnitude proportional to the distance from the center of the sphere. Using this principle, a train starting at rest and traveling in a vacuum without friction on a straight line tunnel from Los Angeles to Tokyo experiences a force in its direction of motion equal to \(-k r \cos \theta\), where • r is the distance of the train from the center of the earth, • x is the distance of the train from the center of the tunnel, • \(\theta\) is the angle between r and x, • 2d is the length of the tunnel between L.A. and Tokyo, • R is the radius of the earth (4,000 miles), as shown in Fig. 4.1.15. ________________ Equation Transcription: Text Transcription: -kr cos theta theta

Factoring Out Friction The damped oscillator equation (2) can be solved by a change of variable that "factors out the damping." Specifically, let \(x(t)=e^{-(b / 2 m) t} x(t)\). (a) Show the X(t) satisfies \(m \ddot{X}+\left(k-\frac{b^{2}}{4 m}\right) X=0\) (b) Assuming that \(k-b^{2} / 4 m>0\), solve equation (21 ) for X(t); then show that the solution of equation (2) is \(x(t)=A e^{-(b / 2 m) t} \cos \left(\omega_{0} t-\delta\right)\), where \(\omega_{0}=\sqrt{4 m k-b^{2}} / 2 m\) ________________ Equation Transcription: x(t)=e-(b/2m)t X(t) k -b2/4m > 0 x(t)=Ae-(b/2m)t cos() Text Transcription: x(t)=e^-(b/2m)t X(t) m double dot X + (k-b^2/4m)X=0 k -b^2/4m > 0 x(t)=A^e-(b/2m)t cos(omega_0t - delta) omega_0=sqrt 4mk-b^2/2m

Suggested Journal Entry With the help of equation (22) from Problem 69, describe the different short- and long-term behavior; of solutions of the mass-spring system in the damped and undamped cases. illustrate with sketches.