Somewhere DEEP BELOW THE EARTH's surface, at an UNKNOWN displacement from the Earth's center, a particle of mass m is dangled from a long string, length L; the particle oscillates along a small arc according to the differential equation d^2x/dt^2=-(pi^2/36)x. Here x refers to an angular displacement measured from the vertical and t refers to time. The particle's mass is given by m=3kg. The length of the string is given by L=5 meters. Whenever the particle arrives at a location of x=(pi/12) radians from the vertical, the particle has no instantaneous speed. On both sides of the vertical, that is, x=(pi/12) radians is repeatedly observed to be a 'turning point' for the particle's periodic motion. 1. Draw a clear FREE-BODY diagram of this particle at some arbitrary point during oscillation, making sure to label variables and constants described above.
Somewhere DEEP BELOW THE EARTH's surface, at an UNKNOWN displacement from the Earth's center, a particle of mass m is dangled from a long string, length L; the particle oscillates along a small arc according to the differential equation d^2x/dt^2=-(pi^2/36)x. Here x refers to an angular displacement measured from the vertical and t refers to time.
The particle's mass is given by m=3kg. The length of the string is given by L=5 meters.
Whenever the particle arrives at a location of x=(pi/12) radians from the vertical, the particle has no instantaneous speed. On both sides of the vertical, that is, x=(pi/12) radians is repeatedly observed to be a 'turning point' for the particle's periodic motion.
1. Draw a clear FREE-BODY diagram of this particle at some arbitrary point during oscillation, making sure to label variables and constants described above.
2. Approximating to three significant digits if necessary, what is the angular frequency of this oscillator on a string?
3. Approximating to three significant digits if necessary, how many cycles per second should we expect of this pendulum?
4. What is the particle's approximate SPEED at t=T/4 seconds? (for which T stands for 'Period" of this pendulum).
If your F-B-D (1, above) is correct, you should find it reasonable to believe that: mg sinx=ma
5. Given this, show how angular frequency depends on both the length of the string and the free-fall acceleration constant due to gravity.
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