Periodic Motion IIPeriodic Motion II"And your head is shaking
and your arms are shaking
and your feet are shaking,
cause the Earth is shaking"
- REM
Lecture Outline:
The simple pendulum also exhibits periodic motion. It a mass is tied to the end of a string of length L and allowed to swing back and forth it will exhibit simple harmonic motion. The forces on the mass are show below.
The net force is directed along the arc made by the ball.
The net force is
For small angles we can approximate
Which gives
where x is the displacement along the arc length from the center.
Comparing this to
we can see that the pendulum will exhibit simple harmonic motion in x with w2 = (g/L).
So for small angular displacements the equation of motion for a pendulum can be written as a simple harmonic oscillator equation. For the simple pendulum, this yields an expression for the period of the oscillation of the pendulum:
Notice that the period does not depend on the mass of the pendulum.
A common use of the pendulum is a pendulum clock. The pictures below show how the pendulum is used in a clock.
The clock is powered by a weight that turns the escape wheel, which is connected via a series of gears to the clock hands. The escape wheel moves in precise steps controlled by the swing of the pendulum. As the pendulum swings it rocks the anchor so that the pallets alternately engage the teeth on the gear wheel. Each swing releases the escape wheel for a short interval to allow it to move on by one tooth. As the teeth of the escape wheel move they push the anchor to keep the pendulum swinging. The pendulum is designed to swing at a given period. In most grandfather clocks this period is 2 seconds for obvious reasons.
A. A pendulum grandfather clock runs slow. In order to fix the clock the pendulum should be
If frictional forces are large the amplitude of the oscillations of a SHO will diminish over time. This is called damping. The period of the oscillation is not affected but the amplitude of the oscillation decreases. If the frictional forces are large enough the SHO will not oscillate at all; this is known as overdamping.
A SHO can also have a driving force acting on it the increases the amplitude. When the harmonic oscillator feels a periodic force it will increase in amplitude if the period of the force is the same as the natural period of the SHO. If the periods do not match the amplitude will not increase over time as the force will on average do zero work on the system. But if the periods match the force will do positive work on the SHO adding energy to it. This effect is call resonance. If the force is applied long enough and is strong enough the SHO may exceed its elastic limit. The classic examples of this are the Tacoma Narrows Bridge Collapse and the shattering of a glass with a loud sound at the natural frequency. You can view a short (but large) MPEG video clip of the Tacoma Narrows Bridge collapse.
A. The answer is 2. If the clock runs slow then the interval between movements of the second hand is too long. To fix this we wish to reduce the period of the pendulum swing. The period depends on the square root of the length so we want to shorten the pendulum.
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