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Oscillation of pendulum, inertia, amplitude, wave speed

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6. A physical pendulum consists of a uniform rod of mass M and length L. The pendulum is pivoted at a point that is a distance x from the center of the rod, so the period for oscillation of the pendulum depends on x:
T(x).

(a) What value of x gives the maximum value for T? (Use any variable or symbol stated above as necessary.)
x =

(b) What value of x gives the minimum value for T? (Use any variable or symbol stated above as necessary.)
x =

7. A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M and length L that is pivoted freely about one end, with a solid sphere of the same mass, M, and a radius of L/2 centered about the free end of the rod?

(a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of M and L. (Use any variable or symbol stated above along with the following as necessary: g.)
I =

(b) Obtain an expression for the period of the pendulum for small oscillations. (Use any variable or symbol stated above along with the following as necessary: g.)
T =

(c) Determine the length L that gives a period of T = 3.1 s.

8. A 3-kg mass attached to a spring with k = 157 N/m is oscillating in a vat of oil, which damps the oscillations.
(a) If the damping constant of the oil is b = 14 kg/s, how long will it take the amplitude of the oscillations to decrease to 1% of its original value?

(b) What should the damping constant be to reduce the amplitude of the oscillations by 97% in 3 s?

13. Consider a guitar string stretching 76.0 cm between its anchored ends. The string is tuned to play middle C, with a frequency of 256 Hz, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced 2.08 mm at its midpoint and released to produce this note, what are the wave speed, v, and the maximum speed,
Vmax, of the midpoint of the string?

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6.

A physical pendulum consists of a uniform rod of mass M and length L. The pendulum is pivoted at a point that is a distance x from the center of the rod, so the period for oscillation of the pendulum depends on x:

(a) What value of x gives the maximum value for T? (Use any variable or symbol stated above as necessary.)

Period of such a rod pendulum is given by (1) [1]

(1)

Where is the moment of inertia of a rod about the support (pivot) point which is obtained from the parallel axis theorem

The moment of inertia of the pendulum rod of length , mass at centre of mass
(taken from standard results)

Thus we can say
(2)

(2) in (1) becomes

(3)

Or expressed as

(3)

One can see that the period is dependant on and being the dominant term so maximum period will be reached when takes on its greatest value. As represents the distance from the centre of the bar pendulum, length the maximum value of is

Thus maximum period when

(b) What value of x gives the minimum value for T? (Use any variable or symbol stated above as necessary.)

To find the minimum value of we need to differentiate (3) with respect to and equate it to zero

(4)

Simplifying (4) and equating the numerator to zero we get

When

When

7.

A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass M and length L that is pivoted freely about one end, with a solid sphere of the same mass, M, and a radius of L/2 centered about the free end of the rod. ...

Solution Summary

The oscillation of pendulum, inertia, amplitudes and wave speed are determined.

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