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By learning Physics Topics, we can gain a deeper appreciation for the natural world and our place in it.
What is the Time Taken by a Pendulum to Complete One Oscillation Called?
A simple pendulum is nothing but a small, heavy body suspended from a rigid support with the help of a long string [Fig.]. The heavy body remains in its lowest position A
when the string is vertical. This position OA is called the position of equilibrium of the pendulum. The heavy body is called the pendulum bob. The support (O) from which the bob is suspended is called the point of suspension. The centre of gravity of the suspended bob is the point of oscillation. When the bob is displaced from its equilibrium position by a little distance and then released, the pendulum oscillates about its equilibrium position on either side.
For convenience in the mathematical treatment of the properties of a pendulum, an ideal simple pendulum is considered. A simple pendulum will be ideal if
- the string is weightless,
- the string is inextensible
- no frictional resistance acts on the bob during its oscillation and
- the bob is a point mass.
Conforming to all the conditions stated above is not practically possible; we never get an ideal simple pendulum. Hence, for laboratory use, a small, heavy metal ball is tied to one end of a long, light string and the
system is suspended from a rigid support.
Definition: If a small heavy body, suspended from a rigid support by a long, weightless and inextensible string, can be set into oscillation, then the arrangement is called a simple pendulum.
When the pendulum bob is slightly displaced from the position A to B and then released, it starts oscillating along arc BAC and CAB. This means that the position of the pendulum periodically changes from OB to OC. This to and fro oscillatory motion is periodic. But practical experience shows that due to air resistance and friction at the suspension point, this oscillation slowly subsides and ultimately the pendulum comes to rest along OA, its equilibrium position.
A Few Definitions Related to Simple Pendulum
Plane of oscillation: In the given diagram [Fig.], the straight lines OA, OB and OC lie on the same vertical plane. The pendulum does not leave that plane during oscillation. This plane is called the plane of oscillation.
Effective Length: The distance of the centre of gravity of the bob from the point of suspension is called the effective length of the pendulum. in the case of a spherical bob of radius r, the centre of gravity lies at the centre of the sphere and if the length of the string is 1, then the effective length
L = l + r.
Amplitude and angular amplitude: The maximum displacement of the pendulum bob on either side of its equilibrium position is called amplitude. In the given diagram, AB or AC is the amplitude of the pendulum.
The angle subtended at the point of suspension by the equilibrium position and the maximum displaced position of the pendulum bob is the angular amplitude of the pendulum. In Fig., the angular amplitude = ∠AOB = ∠AOC = θ.
It is to be noted that, the angular amplitude should be less than 40 so that the arc CAB is almost a straight line.
Complete oscillation and period of oscillation or time period: Starting from an end point, when the pendulum bob again returns to the same point, the pendulum completes one complete oscillation. Referring to Fig., the bob starting from the point B reaches C and returns to B. This completes one complete oscillation. During one complete oscillation, the pendulum bob covers twice the total path of its movement. Hence, let us assume that the bob starts from point A towards point B. After reaching point B, the bob starts moving in the opposite direction, crosses A and reaches point C, then from C the bob reaches A. A complete oscillation is executed in this manner also. After executing one complete oscillation the pendulum returns to its initial phase.
The time taken by a pendulum to complete one oscillation is called the period of oscillation or the time period. In other words, time period is the minimum time taken by the pendulum to return to Its starting phase.
The movement from B to C is a half oscillation of the pendulum and the time required for it is called the halftime period or half the period of oscillation.
Frequency: The number of complete oscillations executed in one second by a pendulum is its frequency. If the time period of a pendulum is T, then as per definition, the number of complete oscillations in time T = 1. Hence, in unit time, the number of complete oscillations = \(\frac{1}{T}\)
No from definition, the frequency n = \(\frac{1}{T}\).
The unit of frequency is s-1 or hertz or Hz; the dimension is T-1.
Motion of a Simple Pendulum
A simple pendulum of effective lengths L is oscillating with an angular amplitude not exceeding 4°. The bob of the pendulum is oscillating from B to C on either side of its position of equilibrium O [Fig.].
Let at any instant of motion, the bob of mass m be at P and its displacement from the position of equilibrium OP = x. If the angular displacement is θ, then θ = \(\frac{x}{L}\)rad, provided θ is small and sinθ ≈ θ ≈ tanθ.
At P, the weight mg of the bob acts vertically downwards. The component mg sin≈ tries to bring the bob to the position of equilibrium. As this force acts in a direction opposite to that of displacement, it is the restoring force, expressed as
F = -mgsinθ
= -mgθ [since θ is less than 4°]
= -mg\(\frac{x}{L}\)
Now, the acceleration of the bob,
a = \(\frac{F}{m}\) = \(\frac{-g}{L}\) = -ω2x [where, ω = \(\sqrt{\frac{g}{L}}\)]
As the motion of the bob obeys the equation, a = -ω2x, it can be said that the motion of a simple pendulum with an angular amplitude less than 4° is simple harmonic.
Time period: Time period of the pendulum,
T = \(\frac{2 \pi}{\omega}\) = \(\frac{2 \pi}{\sqrt{\frac{g}{L}}}\)
= 2π\(\sqrt{\frac{L}{g}}\) …. (1)
Mechanical energy of the pendulum: The kinetic energy of the pendulum,
K = \(\frac{1}{2}\)mω2(A2 – x2) = \(\frac{1}{2}\)m\(\frac{g}{L}\)(A2 – x2)
The potential energy of the pendulum,
U = \(\frac{1}{2}\)mω2x2 = \(\frac{1}{2}\)m\(\frac{g A^2}{L}\) = \(\frac{m g A^2}{2 L}\)
Thus if the angular amplitude is less than 4°, total mechanical energy of a simple pendulum is
- directly proportional to the mass of the bob,
- inversely proportional to the effective length of the pendulum and
- directly proportional to the square of the amplitude of the SRM executed by the pendulum.
Tension on the string: When the pendulum oscillates about its point of suspension, a centripetal force is required for the circular motion of the bob. At the instant when the bob passes through the position of equilibrium O, its velocity becomes maximum. So, the upward centripetal force, F, becomes maximum at O. The resultant of downward weight mg and tension F’ on the string becomes equal to this force F.
Therefore, F = F’ – mg or, F’ = Fc + mg
Kinetic energy of the pendulum at P,
When the bob passes through O (where x = 0), the tension F’ on the string becomes maximum.
∴ F’max = mg(1 + \(\frac{A^2}{L^2}\))