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# Exercises

1. If a train of mass is subject to a retarding force , show that if the engines are shut off when the speed is then the train will come to rest in a time after traveling a distance 2. A particle is projected vertically upward from the Earth's surface with a velocity which would, if gravity were uniform, carry it to a height . Show that if the variation of gravity with height is allowed for, but the resistance of air is neglected, then the height reached will be greater by , where is the Earth's radius.

3. A particle is projected vertically upward from the Earth's surface with a velocity just sufficient for it to reach infinity (neglecting air resistance). Prove that the time needed to reach a height is where is the Earth's radius, and its surface gravitational acceleration.

4. A particle of mass is constrained to move in one dimension such that its instantaneous displacement is . The particle is released at rest from , and is subject to a force of the form . Show that the time required for the particle to reach the origin is 5. A block of mass slides along a horizontal surface which is lubricated with heavy oil such that the block suffers a viscous retarding force of the form where is a constant, and is the block's instantaneous velocity. If the initial speed is at time , find and the displacement as functions of time . Also find as a function of . Show that for the block does not travel further than .

6. A particle is projected vertically upward in a constant gravitational field with an initial speed . Show that if there is a retarding force proportional to the square of the speed then the speed of the particle when it returns to the initial position is where is the terminal speed.

7. A particle of mass moves (in one dimension) in a medium under the influence of a retarding force of the form , where is the particle speed, and and are positive constants. Show that for any value of the initial speed the particle will never move a distance greater than , and will only come to rest as .

8. Two light springs have spring constants and , respectively, and are used in a vertical orientation to support an object of mass . Show that the angular frequency of oscillation is if the springs are in parallel, and if the springs are in series.

9. A body of uniform cross-sectional area and mass density floats in a liquid of density (where ), and at equilibrium displaces a volume . Show that the period of small oscillations about the equilibrium position is 10. Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.

11. If the amplitude of a damped harmonic oscillator decreases to of its initial value after periods show that the ratio of the period of oscillation to the period of the same oscillator with no damping is 12. Consider a damped driven oscillator whose equation of motion is Let and at .
1. Find the solution for when .
2. Find the solution for when .

13. Obtain the time asymptotic response of a damped linear oscillator of natural frequency and damping coefficient to a square-wave periodic forcing function of amplitude and frequency . Thus, for , , etc., and for , , etc.   Next: Multi-Dimensional Motion Up: One-Dimensional Motion Previous: Simple Pendulum
Richard Fitzpatrick 2011-03-31