Applications of Linear Equations with Constant Coefficients 289EXERCISES 6.1.1
Exercises 1-9 pertain to simple pendulums. In these exercises
assume the angular displacement, y, is small enough so that the
approximation sin y,...., y is valid and use the linearized initial value
problem (2) to describe the motion of the pendulum. On the earth
g = 9.8 meters/second^2 (m/s^2 ) is the mean surface gravitational
constant and on the moon g = 1.62 m/s^2 •
l. A child whose mass is 40 kilograms (kg) is swinging in a park on a swing
of length 5 meters (m). Assuming air resistance is negligible, calculate
the period and frequency of oscillation.- If an undamped pendulum on the earth has a period of 1 second, how
long is the pendulum in meters? How long must a pendulum be in order
to have a period of 2 seconds? How long would an undamped pendulum
need to be on the moon to have a period of 1 second? - Does an undamped pendulum of length f, with a .2 kg bob oscillate
faster or slower than an undamped pendulum of length f with a .1 kg
bob? If both pendulums are subjected to the same damping, which will
oscillate faster? - Two identical undamped pendulums, one on the earth and the other on
the moon, are started with the same initial displacement and velocity.
Which pendulum will oscillate faster? Which pendulum will have the
larger ampli tude? - An undamped pendulum of length .5 m with a bob of mass .3 kg is
moved to the right of vertical so that the pendulum makes an angle of
1 radian with the vertical and is then released without imparting any
velocity to the pendulum.
a. Write the equation of motion for the pendulum.
b. What is the amplitude and period of oscillation?
c. What is the phase angle?
d. What is the velo city and acceleration of the pendulum when the
bob is at the vertical and headed toward the left? right?- An undamped pendulum of length 1/6 m with a bob of mass .2 kg is
started with an angular displacement of .5 radians and released with an
angular velocity of 1 radian/second.
a. Write the equation of motion for the pendulum.
b. What is the amplitude and period of oscillation?