PHY183-Lecture47 - Review: SHM ! Hooke's Force Law: F = !kx...

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April 14, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 47 April 14, 2006 Physics for Scientists&Engineers 1 2 Review: SHM Review: SHM ! Hooke’s Force Law: ! Equation of motion: ! Solution: or: ! Amplitude: Phase: ! Angular Frequency: F = ! kx d 2 x dt 2 + k m x = 0 x ( t ) = A sin( ! 0 t ) + B cos( 0 t ) x ( t ) = C sin( 0 t + " 0 ) C = A 2 + B 2 0 = arctan( B / A ) 0 = k m m d 2 x dt 2 = ! kx April 14, 2006 Physics for Scientists&Engineers 1 3 Review: SHM (2) ! Period: ! Frequency: T = 2 0 = 2 k / m = 2 m k f = 0 2 = 1 2 k m April 14, 2006 Physics for Scientists&Engineers 1 4 Work and Energy in Harmonic Oscillations ! We have calculated the functions v(t) and x(t) that describe the oscillation of a mass on a spring in time ! We can use these to find the potential energy stored in a spring as a function of time ! And the kinetic energy as a function of time where we have used U = 1 2 kx 2 = 1 2 kC 2 sin 2 ( 0 t + # 0 ) K = 1 2 mv 2 = 1 2 m 0 2 C 2 cos 2 ( 0 t + 0 ) = 1 2 kC 2 cos 2 ( 0 t + 0 ) 0 2 = k m x ( t ) = C sin( 0 t + 0 ) v ( t ) = 0 C cos( 0 t + 0 )
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April 14, 2006 Physics for Scientists&Engineers 1 5 Work and Energy in Harmonic Oscillations (4) ! Combining these we find the total Energy: using ! The total Energy is constant, as found previously. ! It oscillates between K(t) and U(t). E = K + U = 1 2 kC 2 cos 2 ( " 0 t + # 0 ) + 1 2 kC 2 sin 2 ( 0 t + 0 ) = 1 2 kC 2 cos 2 + sin 2 = 1 April 14, 2006 Physics for Scientists&Engineers 1 6 Quiz 46.1 ! The driver of a car decides to smoke a cigar. ! Which window should be opened so that the passenger, sitting beside the driver, does not have to breathe the secondhand smoke? A) The driver-side window B) The passenger-side window April 14, 2006 Physics for Scientists&Engineers 1 7 Pendulum Motion Pendulum Motion ! We have asserted that the angle !
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PHY183-Lecture47 - Review: SHM ! Hooke's Force Law: F = !kx...

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