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Download Sects. 12.3 through 12.4
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Chapter 12 – part B The Energy of an Harmonic Oscillator The Pendulum Exercise 12.14 14. A 200-g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. The total energy of the system is 2.00 J. Find (a) the force constant of the spring and (b) the amplitude of the motion. 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 12.18 18. A 2.00-kg object is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest with an initial position of xi = 0.200 m, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the object. Where does this maximum speed occur? (d) Find the maximum acceleration of the object. Where does it occur? (e) Find the total energy of the oscillating system. Find (f) the speed and (g) the acceleration of the object when its position is equal to one third of the maximum value. 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 12.24 24. The angular position of a pendulum is represented by the equation θ = (0.032 0 rad) cos ωt, where θ is in radians and ω = 4.43 rad/s. Determine the period and length of the pendulum. Exercise 12.25 25. A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if it starts from rest with a small displacement from equilibrium, the particle moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R (that is, ). Exercise 12.47 47. A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension. Find the frequency of vibration of the system for small values of the amplitude (small θ). Assume that the vertical suspension of length L is rigid, but ignore its mass.