Transcript Sects. 12.1 through 12.2
Chapter 12 – part A The kinematics and dynamics of the Simple Harmonic Oscillator Summary Kinematics The motion Dynamics Important relations T A Other important quantities Exercise 12.4 4. A particle moves in simple harmonic motion with a frequency of 3.00 Hz and an amplitude of 5.00 cm. (a) Through what total distance does the particle move during one cycle of its motion? (b) What is its maximum speed? Where does this maximum speed occur? (c) Find the maximum acceleration of the particle. Where in the motion does the maximum acceleration occur? Exercise 12.6 6. A particle moves along the x axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration –0.320 m/s2. First, assume that it moves with constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume that it moves with simple harmonic motion for 4.50 s and that x = 0 is its equilibrium position. Find (c) its position and (d) its velocity at the end of this time interval. Exercise 12.8 8. A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find (a) the period of its motion, (b) the frequency in hertz, and (c) the angular frequency in radians per second. Exercise 12.11 11. A 0.500-kg object attached to a spring with a force constant of 8.00 N/m vibrates in simple harmonic motion with an amplitude of 10.0 cm. Calculate (a) the maximum value of its speed and acceleration, (b) the speed and acceleration when the object is 6.00 cm from the equilibrium position, and (c) the time interval required for the object to move from x = 0 to x = 8.00 cm. Exercise 12.12 12. A 1.00-kg glider attached to a spring with a force constant of 25.0 N/m oscillates on a horizontal, frictionless air track. At t = 0, the glider is released from rest at x = –3.00 cm (that is, the spring is compressed by 3.00 cm). Find (a) the period of its motion, (b) the maximum values of its speed and acceleration, and (c) the position, velocity, and acceleration as functions of time. 77. Figure P10.77 shows a vertical force applied tangentially to a uniform cylinder of weight Fg. The coefficient of static friction between the cylinder and all surfaces is 0.500. In terms of Fg, find the maximum force P that can be applied that does not cause the cylinder to rotate. (Suggestion: When the cylinder is on the verge of slipping, both friction forces are at their maximum values. Why?) 76. A solid sphere of mass m and radius r rolls without slipping along the track shown in Figure P10.76. It starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R, much larger than r. (a) What is the minimum value of h (in terms of R) such that the sphere completes the loop? (b) What are the force components on the sphere at the point P if h = 3R?