Please
Do Not Use This Reference Material WORD FOR WORD View Terms Of usage policy Page of this site for more Info on how to use this site.
Title: Compound Pendulum - Conditions For Minimum Period
Aim:
Please
Do Not Use This Reference Material WORD FOR WORD View Terms Of usage policy Page of this site for more Info on how to use this site.
THEORY:
GENERAL
The length L of the ideal simple pendulum
discussed above is the distance from the pivot
point to the center of
mass of the bob. Any swinging rigid body free to rotate
about a fixed horizontal axis is called a compound pendulum or physical
pendulum. The appropriate equivalent length L for calculating the
period of any such pendulum is the distance from the pivot to the center of
oscillation. This point is located under
the center of mass
at a distance from the pivot traditionally called the radius of oscillation,
which depends on the mass distribution of the pendulum. If most of the mass is
concentrated in a relatively small bob compared to the pendulum length, the
center of oscillation is close to the center of mass.
The radius of oscillation or equivalent length L
of any physical pendulum can be shown to be
where I is the moment of inertia
of the pendulum about the pivot point, m is the mass of the pendulum,
and R is the distance between the pivot point and the center of mass.
Substituting this expression in (1) above, the period T of a compound
pendulum is given by
for sufficiently small oscillation.
A rigid uniform rod of length L pivoted
about either end has moment of inertia I = (1/3)mL2.
The center of mass is located at the center of the rod, so R = L/2.
Substituting these values into the above equation gives T = 2π√2L/3g. This shows that a rigid rod pendulum
has the same period as a simple pendulum of 2/3 its length.
THEORY
1
The compound pendulum
A simple pendulum theoretically has the mass of the
bob concentrated at one point, but this is impossible to achieve exactly in
practice. Most pendulums are compound, with an oscillating mass spread
out over a definite volume of space.
Let G be the centre of gravity of a compound pendulum of mass m that oscillates about a point O with OG = h If the pendulum is moved so that the line OG is displaced through an angle θ (Figure 1), the restoring couple is:
- mghsinθ = - mghθ = if θ is
small.
Therefore:
I= = - mghθ = and so = - mgθ h /I
Since the angular acceleration is directly proportional to the angular displacement the motion is simple harmonic of period T where:
Period
of a compound pendulum (T) = 2π(I/mgh)1/2
But I is the moment of inertia about an axis through 0, and therefore
I = IG + mh2 = mk2 + mh2
where k is the radius of gyration about a parallel axis through G. The period can therefore be written as:
Period
of a compound pendulum (T) = 2π([k2 + h2]/gh)1/2
If a uniform rod is used as a compound pendulum and
the period of oscillation T measured for different values of h on either side
of the centre of gravity then a graph like the one in Figure 2 may be obtained.
Since the formula for a simple pendulum is T = 2π(L/g)1/2 we can define a quantity L called the length of the simple equivalent pendulum.
This is given by L = [k2 + h2]/h
For two distances h1 and h2 on either side of the centre, L = h1 + h2 (as can be seen from the graph in figure 2) and h1h2 = k2. At the minimum h1 = h2 and h = k. A value of g can be determined by measuring L from the graph.
Since the formula for a simple pendulum is T = 2π(L/g)1/2 we can define a quantity L called the length of the simple equivalent pendulum.
This is given by L = [k2 + h2]/h
For two distances h1 and h2 on either side of the centre, L = h1 + h2 (as can be seen from the graph in figure 2) and h1h2 = k2. At the minimum h1 = h2 and h = k. A value of g can be determined by measuring L from the graph.
A simple pendulum consists of a small body called a “bob” (usually a sphere) attached to the end of a string the length of which is great compared with the dimensions of the bob and the mass of which is negligible in comparison with that of the bob. Under these conditions the mass of the bob may be regarded as concentrated at its center of gravity, and the length l of the pendulum is the distance of this point from the axis of suspension. When a simple pendulum swings through a small arc, it executes linear simple harmonic motion of period T, given by the equation
T = 2π √(l/g) (1)
where g is the acceleration due to gravity. This relation-ship affords one of the simplest and most satisfactory methods of determining g experimentally.
When the dimensions of the suspended body are not negligible in comparison with the distance from the axis of suspension to the centre of gravity, the pendulum is called a compound, or physical, pendulum.Any body mounted upon a horizontal axis so as to vibrate under the force of gravity is a compound pendulum. The motion of such a body is an angular vibration about the axis of suspension. The expression for the period of a compound pendulum may be deduced from the general expression for the period of any angular simple harmonic motion
T = 2π√(− θ/a) (2)
and the application of Newton’s second law of motion for rotating bodies
L = Ia (3)
where θ is the angular displacement, α the angular acceleration, L the torque and I the rotational inertia of the body.
L = mgh sinθ (4)
where h is the distance from the axis of suspension to the center of gravity. If a minus sign is used to indicate the tact that the torque L is opposite in sign to the displacement θ, Eqs. (3) and (4) yield
Ia = −mgh sinθ (5)
When the angular displacement θ is sufficiently small, sinθ is approximately equal to θ measured in radians. With this restriction Eq. (5) may be written
a = ((−mgh)/I)θ (6)
Since m, g, h and I are all numerically constant for any given case, Eq. (6), may be written simply
a = −kθ (7)
where k is a constant. Equation (7) is the defining equation of angular simple harmonic motion, i.e., motion in which the angular acceleration is directly proportional to the angular displacement and oppositely directed. Since the system executes angular simple harmonic motion, substitution of the expression for a from Eq. (6) in Eq. (2) yields the equation for the period of a compound pendulum
T = 2π √(I/mgh) (8)
where I is the rotational inertia of the pendulum about the axis of suspension S. It is convenient to express I in terms of I0, the rotational inertia of the body about an axis through its center of gravity G. If the mass of the body is m,
Io = mko2 (9)
where ko is the radius of gyration about an axis through G. For any regular body, ko can be computed by means of the appropriate formula (see any handbook of physics or engineering); for an irregular body it must be determined experimentally. The rotational inertia about any axis parallel to the one through the center of gravity is given by
I = Io + mh2 (10)
where h is the distance between the two axes. Substitution of the relationships of Eqs. (9) and (10) in Eq. (8) yields
T = 2π √((ko2 + h2)/gh) (11)
This equation expresses the period in terms of the geometry of the body. It shows that the period is independent of the mass, depending only upon the distribution of the mass (as measured by ko) and upon the location of the axis of suspension (as specified by h). Since the radius of gyration of any given body is a constant, the period of any given pendulum is a function of h only. Comparison of Eq. (1) and Eq. (11) shows that the period of a compound pendulum suspended on an axis at a distance h from its center of gravity is equal to the period of a simple pendulum having a length given by
l = (ko2 + h2)/h = h + (ko2/h) (12)
The simple pendulum whose period is the same as that of a given compound pendulum is called the “equivalent simple pendulum.”
It is sometimes convenient to specify the location of the axis of suspension S by its distance s from one end of the bar, instead of by its distance h from the center of gravity G. If the distances s1, s2 and D (Fig. 1) are measured from the end A, the distance h1 must be considered negative, since h is measured from G. Thus, if D is the fixed distance from A to G, s1 = D + h1, s2 = D + h2 and, in general, s = D + h. Substitution of this relationship in Eq. (11) yields
T = 2π √((ko2 + (s − D)2)/(g(s − D))) (13)
A horizontal line SS’, corresponding to a chosen value of T, intersects the, graph in four points, indicating that there are four positions of the axis, two on each side of the center of gravity, for which the periods are the same. These positions are symmetrically located with respect to G. There are, therefore, two numerical values of h for which the period is the same, represented by h1 and h2 (Figs. 1 and 2). Thus, for any chosen axis of suspension S there is a conjugate point O on the opposite side of the center of gravity such that the periods about parallel axes through S and O are equal. The point O is called the center of oscillation with respect to the particular axis of suspension S. Consequently, if the center of oscillation of any compound pendulum is located, the pendulum may be inverted and supported at O without altering its period. This so-called reversibility is one of the unique properties of the compound pendulum and one that has been made the basis of a very precise method of measuring g (Kater’s reversible pendulum).
It can be shown that the distance between S and O is equal to l, the length of the equivalent simple pendulum. Equating the expressions for the squares of the periods about S and O, respectively, from Eq. (11)
T2 = (4π2/g)((ko2 + h12)/h1) = (4π2/g)((ko2 + h22)/h2) (14)
from which
T2 = (4π2/g) (h1 + h2) (15)
or T = 2π√((h1 + h2)/g) (16)
Comparison of Eqs. (1) and (16) shows that the length l of the equivalent simple pendulum is
l = h1 + h2 (17)
Thus, the length of the equivalent simple pendulum is SO (Figs. 1 and 2).
S’ and O’ are a second pair of conjugate points symmetrically located with respect to S and O respectively, i.e., having the same numerical values of h1 and h2.
Further consideration of Fig. 2 reveals the fact that the period of vibration of a given body cannot be less than a certain minimum value To, for which the four points of equal period reduce to two, S and O’ merging into P, and S’ and O merging into P’ as h1 becomes numerically equal to h2. The value of ho corresponding to minimum period can be deduced by solving Eq. (14) for ko2, which yields
ko2 = h1h2 (18)
and setting
ho = h1 = h2 (19)
Thus
ho = ko (20)
Substituting in Eq. (12) yields
lo = 2ko (21)
Thus, the shortest simple pendulum to which the compound pendulum can be made equivalent has a length lo equal to twice the radius of gyration of the body about a parallel axis through the center of gravity. This is indicated in Fig. 2 by the line PP’. Inspection of the figure further shows that, of the two values of h for other than minimum period, one is less than and one greater than ko. From the foregoing it is evident that if two asymmetrical points S and O can be found such that the periods of vibration are identical, the length of the equivalent simple pendulum is the distance between the two points, and the necessity for locating the center of gravity is eliminated. Thus, by making use of the reversible property of the compound pendulum, a simplicity is, achieved similar to that of the simple pendulum, the experimental determinations being reduced to one measurement of length and one of period.
For Procedures, See your Mechanical Laboratory Manual.
Please
Do Not Use This Reference Material WORD FOR WORD View Terms Of usage policy
Page of this site for more Info on how to use this site
Observations :-
1.
Applications:-
1.
PRECAUTIONS:
For Precautions, See General Laboratory Precautions
Please
Do Not Use This Reference Material WORD FOR WORD View Terms Of usage policy Page of this site for more Info on how to use this site.
Blogger Comment
Facebook Comment