Simple Pendulum: Definition, Time Period Derivation, Energy Consideration
Simple Pendulum Definition
A simple pendulum can be defined as a device with a point mass attached to a light, non-stretchable cord and suspended from a fixed support. The vertical line through the fixed support is the center position of the simple pendulum. The vertical distance between the hanging point and the center of gravity of the suspended object (if in the center position) is called the length of the simple pendulum and is represented by L. This form of the pendulum is based on a resonant system with a single resonant frequency.
The pendulum bob swings back and forth when you draw it to one side and release it. It sways back and forth. At this stage, you don’t know whether or not the bob oscillates in a simple harmonic motion, but you do know that it does. Simply calculate whether its acceleration is a negative constant times its position to see if it is subject to simple harmonic motion. Because the bob oscillates in an arc rather than a straight line, angular variables make it easier to evaluate the motion.
The gravitational force is the primary driver of this motion, which occurs in a vertical plane. The bob that is hanging at the end of a thread is extremely light, almost massless. Increase the length of the string while taking measurements from the point of suspension to the middle of the bob to prolong the period of a simple pendulum.
A simple pendulum is a mechanical system that moves in a predictable manner. A little bob of mass ‘m’ is suspended by a thin string attached to a platform at its upper end of length L in the basic pendulum. The simple pendulum is a mechanical mechanism that oscillates.
The time period of Pendulum
The point mass m hangs at the end of the cord, which is light and does not stretch. The top of the cord is attached to a rigid support. The mass moves from the center position. We have to assume negligible friction from air to system, massless string, perfect swing of the pendulum, and no constant gravity influence for the derivation of the time period.
From the equation of motion, we have
T – mg cosθ = mv2L
Where,
m = mass of the bob
v= velocity
L= length of the string
T= tension
g= acceleration due to gravity
The torque tends to bring the mass to its equilibrium position,
τ = mgL × sinθ
= mgsinθ × L
= I × α
For small angles of oscillations,
sin θ ≈ θ,
Therefore, Iα = -mgLθ
α = -(mgLθ)/I
– ω02θ = -(mgLθ)/I
ω02= (mgL)/I
ω02= √(mgL/I)
Using I = MI
Where,
I denote the moment of inertia. a
we get,
ω0 = √(g/L)
Thus, the time period of the pendulum is given as,
T = 2π/ω0 = 2π × √(L/g)
Energy consideration of simple pendulum
Let us consider that the bob is pulled out at a distance of Xmax from the equilibrium position, and then is. The bob already possesses a certain amount of energy before it picks up any speed. As we are dealing with an ideal system, the energy then remains constant.
E=K+U
Where
E = Total Energy
K= Total kinetic Energy
U= Total potential energy
When the bob is oscillating, the energy is partly kinetic, K = 12mv2, and partly potential energy,
U =
Where
k= spring constant
x= distance traveled by the bob
initially, the bob is at rest
So,
E = U = k
Towards the end of the motion, the bob lies at an easy position at which we only calculate the potential energy. As the spring contracts, the bob is pulled towards the wall, and the speed of the block increases but the potential energy
decreases as the spring become less and less stretched. While attaining the equilibrium position, the system has both kinetic and potential energy.
E = K+U
When the bob reaches the equilibrium condition, the Kinetic Energy keeps increasing but the Potential energy becomes zero. Hence, at equilibrium condition,
E = K+0
Or, E =m+ 0
The block continues to move. It starts compressing the spring after it overshoots the equilibrium position. As it compresses the spring, the speed decreases. Spring potential energy is converted from kinetic energy. The constant value of total energy represents a combination of kinetic and potential energy as the block moves closer to the wall, with the kinetic energy decreasing and the potential energy increasing.
The block eventually has a velocity of zero at its closest point of approach to the wall, its maximum displacement in the negative x-direction from its equilibrium position, at its turning point. The kinetic energy is zero at that point, while the potential energy is at its highest value.
E= U
The block then begins to move away from the wall. Its kinetic energy rises as its potential energy falls, bringing it back to the equilibrium state. The spring is neither stretched nor compressed at that moment, hence the potential energy is zero. The entire amount of energy is kinetic energy. The block continues past the equilibrium point due to its inertia, stretching the spring and slowing down as the kinetic energy reduces while the potential energy grows at the same time.
The block eventually returns to its original position, but only for a brief moment, at rest and devoid of kinetic energy. Throughout the oscillatory action, the total energy has remained constant.
References
- https://www.sciencedirect.com/topics/engineering/simple-pendulum
- https://blogs.bu.edu/ggarber/interlace/pendulum/energy-in-a-pendulum/#:~:text=In%20a%20simple%20pendulum%20with,energy%20and%20gravitational%20potential%20energy.
