THE SIMPLE PENDULUM (ODE)

• NAKRANI DARSHAN D (D -17)• PATIL DIPESH J (D-57)• MODI RAHUL Y ( D- 15)

AEM TOPIC:

The Simple Pendulum

A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass).

The angle it makes with the vertical varies with time as a sine or cosine.

The Simple Pendulum

Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sinθ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case).

The Simple Pendulum

Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on the length of the string:

The Simple Pendulum

In this case, it can be shown that the period depends on the moment of inertia:

Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,

Example, pendulum:

In Fig. a, a meter stick swings about a pivot point at one end, at distance h from the stick’s center of mass.(a)What is the period of oscillation T?

KEY IDEA: The stick is not a simple pendulum becauseits mass is not concentrated in a bob at the end oppositethe pivot point—so the stick is a physical pendulum.

Calculations: The period for a physical pendulum depends on the rotational inertia, I, of the stick about the pivot point. We can treat the stick as a uniform rod of length L and mass m. Then I =1/3 mL2, where the distance h is L. Therefore, ti

Note the result is independent of the pendulum’s mass m.

Simple Harmonic Motion (SHM). The simple pendulum.

• Calculate the angular frequency of the SHM of a simple pendulum.– A simple pendulum is a

pendulum for which all the mass is located at a single point at the end of a massless string.

– There are two forces acting on the mass: the tension T and the gravitational force mg.

– The tension T cancels the radial component of the gravitational force.

Example, pendulum, continued:(b) What is the distance L0 between the pivot point O of the stick and the center of oscillation of the stick?

Calculations: We want the length L0 of the simple pendulum (drawn in Fig. b) that has the same period as the physical pendulum (the stick) of Fig. a.

Simple Harmonic Motion

The time to complete one full cycle of oscillation is a Period.

T 1f

f 1T

The amount of oscillations per second is called

frequency and is measured in Hertz.

Simple Harmonic Motion

An objects maximum displacement from its equilibrium

position is called the Amplitude (A) of the motion.

kmTPeriod 2

x(t) Acos

ddt

t

x(t) Acos t

Start with the x-component of position of the particle in UCM

End with the same result as the spring in SHM!

Notice it started at angle zero

Initial conditions:

t 0

We will not always start our clocks at one amplitude.

x(t) Acos t 0

Acceleration is at a maximum when the particle is at maximum and minimum displacement from x=0.

tAdt

tAddt

tdva xx

cos

sin

)(

2

Acceleration is proportional to the negative of the

displacement.

ax 2Acos t

ax 2x

x Acos t

As we found with energy considerations:

ax 2x

F ma x kx

max kx

ax km

x

According to Newton’s 2nd Law:

ax d2xdt2

Acceleration is not constant:

d2xdt 2

km

x

This is the equation of motion for a mass on a spring. It is of a

general form called differential equation.

Differential Equations:

d2xdt 2

km

x

IT WORKS. Sinusoidal oscillation of SHM is a result of Newton’s laws!

x Acos t 0

d2xdt 2 2Acos t

dxdt

Asin t

2Acos t km

Acos t

2 km

• we get the two graphs below. Showing the difference between the simple harmonic model and the small angle approximation model.

Assumptions• All models are full of assumptions. Some of these assumptions are very accurate,

such as the pendulum is una ected by the day of the week. Some of these ffassumptions are less accurate but we are still going to make them, friction does not e ect the system. Here is a list of some of the more notable assumptions of this ffmodel of a pendulum.

• Friction from both air resistance and the system is negligible.• The pendulum swings in a perfect plane.• The arm of the pendulum cannot bend or stretch/compress.• The arm is mass less.• Gravity is a constant 9.8 meter/second2.

Applications• Pendulums have many applications and were utilized often before the digital age.

They are used in clocks and metronomes due to the regularity of their period, in wrecking balls and playground swings, due to their simple way of building up and keeping energy.

Conclusion

• A pendulum is easy to make and with a little bit of math, easy to understand, one could even use the swaying of their hammock, assuming a fairly uniform driving force.

Reference

• The Simple Pendulum www.acs.psu.edu/drussell/Demos/

• Pendulum (mathematics) www.wikipedia.org• Mathematical Swingers: The Simple Pendulum

as a Log Application www.http://my.execpc.com.

• R.S.KHURMI PUBLICATION (Theory of Machine CH-4)

Thank You