Note to problem solvers:simple harmonic
motion is the most complicated motion we will study all year.
Simple harmonic oscillators move with variable acceleration such
that when the acceleration is a maximum. velocity is 0; and when
the velocity is a maximum, the acceleration is 0. See page 7 of
the equations page for a listing of handy tools and relationships
that apply to this topic
1. A pendulum has a length of 1 m. What is its period? What is its frequency? Let g = 9.80 m/s/s. What happens to the period if the length is doubled? If the mass of the pendulum is doubled? If the initial displacement of the pendulum is doubled?
2. A spring scale can be stretched .1 m when a weight of 40 N is attached to it. A mass of 2.6 kg is attached to the spring and released. What are the period and frequency if oscillation?
3. A seconds pendulum has a period of exactly 1 s. How long is it? (Assume g = 9.80 m/s/s.)
4. A pendulum is used to drive a grandfather clock. Assume that the clock is calibrated for a site on the Earth where g is exactly 9.8000 m/s/s. By how much does the clock gain or lose in a month if the clock is moved to a place where g = 9.8100 m/s/s? By how much does a clock gain or lose in month if thermal conditions increased the length of the pendulum by .01 percent?
5. A mass (m = 2 kg) is attached to a spring k = 500 N/m. The mass is displaced 45 cm from the equilibrium and then released.a) What is the maximum velocity of the mass? Where did this occur? What is the maximum acceleration? Where does this occur? c) What are the speed and acceleration when the displacement x = 25 cm?
6. Same oscillator as #5 above. Determine X, V & A when t = .7 s.
7. a simple harmonic oscillator has an amplitude of 40 cm and moves with a period of three seconds. Determine the displacement, velocity, and acceleration of this object when t = 1 second. Determine how fast the object is moving and its acceleration when its displacement is +30 cm.
8. a platform moves vertically as a simple harmonic oscillator wit a period of 2 seconds. A 2 kg block is placed on the platform. What must be the maximum amplitude that the oscillator could have without the block leaving the platform at any time?
9. A block has a mass of one kg and is attached to a spring whose force constant equals 1000 newtons/meter. A gun having a muzzle velocity of 200 m/s fires a bullet (mass is 50 grams) into the block. A) by how much does the spring compress? B) The block and spring subsequently exhibit simple harmonic motion, what is a period of vibration of this oscillator? What is its maximum velocity? What is its maximum acceleration?
10. astronauts deployed to the space station will have to monitor their weight for possible gain or loss. The problem is that in the micro gravity that exists around the earth, how do they weigh themselves? The answer is a device called the inertial balance. Such a device is essentially a spring with a chair fastened to it. The astronauts sit in the chair and cause the system to vibrate back and forth. If one knows a spring constant for the spring and measures the period of the spring, in principal we can determine the mass of the astronauts. Let us say on the first any of the voyage, our astronaut had a mass of 80 kgs and caused the inertial balance of vibrating with frequency of .1 vibrations per second. Five days later under similar circumstances the mechanism vibrates with a frequency of .09 vibrations/second. By how much has the mass of the astronauts changed?