V. Energy Conservation in Oscillatory Motion
Chapter 13. Oscillations About Equilibrium
I. Periodic Motion
1. A child on a merry-go-around completes a full rotation in 30 s. What is the frequency of rotation? 2. Convert 55 rev/min (revolutions per minute) to Hz. 3. The angular frequency of a point on a rotating wheel is 100 rev/min (revolutions per minute). What is this angular velocity in rad/s? 4. The angular frequency of a point on a rotating wheel is 15 rad/s. What is the period of rotation of that point?
1. The position of an oscillating object is described by the following equation: x = 3cos(3t). What is the amplitude (frequency, angular frequency, period) of the oscillation?
III. Connections Between Uniform Circular Motion and Simple Harmonic Motion
1. The acceleration of a body in simple harmonic motion is represented by the formula a = - 5 cos (ωt), where a is the acceleration and ω is the angular frequency. Knowing that the amplitude of oscillation is 1 m, find the period (angular frequency, frequency, maximum speed).
IV. The Period of a Mass on a Spring
1. A 1-kg mass is attached to a vertical spring. Knowing that the force constant of the spring equals 300 N/m and that the maximum acceleration of the mass is 10 m/s2, find the amplitude of oscillation.
V. Energy Conservation in Oscillatory Motion
1. A spring positioned vertically with a 1-kg mass attached to its end is stretched by ∆x = 10 cm past the equilibrium position. At time t = 0 s the mass is released. Find the potential energy of the mass 12 s later if the force constant of the spring is 400 N/m.
VI. The Pendulum
1. An odd-shaped 4-kg physical pendulum is oscillating with a period of 6 s. What is the distance between the pivot point and the center of mass of the pendulum if its moment of inertia is 40 kg·m2? Assume the acceleration due to gravity to be 9.81 m/s2.