A pendulum is formed by pivoting a long thin rod about a point on the rod. In a series of experiments, the period is measured as a function of the distance x between the pivot point and the rod’s center.
The problem deals with the calculation of moment of inertia. It is the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation. Using the equation for moment of inertia of rod about its center and the parallel axis theorem, we can calculate the moment of inertia of the rod about the pivot point.
Where T is time, m is mass L is the length, I is the rotational inertia, and is the rotational inertia about the center
The equation for rotational inertia of a rod about its center is
Now, according to the parallel axis theorem, the rotational inertia of a rod about a point distance ‘d’ from its center is
Now, the equation for period of the pendulum is
By substituting the equation for moment of inertia in this equation, we get
Now, to get the minimum period, we differentiate this equation with respect to x
By solving this equation, we get
Now, substituting this equation, we can get Tmin as,
We have derived the equation for minimum period as
As, the term L is in the numerator, the period will increase with increase in length of pendulum.
The equation for minimum period is
From this equation, we can conclude that there is no effect of mass on the period as there is no mass term in this equation so that the period will remain same
The vibration frequencies of atoms in solids at normal temperatures are of the order of. Imagine the atoms to be connected to one another by springs. Suppose that a single silver atom in a solid vibrates with this frequency and that all the other atoms are at rest. Compute the effective spring constant. One mole of silver (atoms) has a mass of 108 g.
A grandfather clock has a pendulum that consists of a thin brass disk of radius r = 15.00 cm and mass 1.000 kg that is attached to a long thin rod of negligible mass. The pendulum swings freely about an axis perpendicular to the rod and through the end of the rod opposite the disk, as shown in Fig. 15-5 If the pendulum is to have a period of 2.000 s for small oscillations at a place where ,what must be the rod length L to the nearest tenth of a millimeter?
The velocity of a particle undergoing SHM is graphed in Fig. . Is the particle momentarily stationary, headed toward, or headed toward at (a) point A on the graph and (b) point B? Is the particle at, at, at 0, between and 0, or between 0 andlocalid="1657280889199" when its velocity is represented by (c) point A and (d) point B? Is the speed of the particle increasing or decreasing at (e) point A and (f) point B?
Question: In Figure, the pendulum consists of a uniform disk with radius r = 10.cm and mass 500 gm attached to a uniform rod with length L =500mm and mass 270gm.
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