Simple Pendulum, practice problem, FAQs

If you remember, in our childhood we tied rope to a stone and oscillated in front of someone and said I am hypnotizing you. What is this rope and stone system? Similarly in your physics lab you had seen a metal ball suspended with a string. These are examples of a simple pendulum. Lets Know more about simple pendulums.

Table of content

• Simple pendulum
• Seconds pendulum
• Simple pendulum in accelerating frame
• Practice problem
• FAQs

Simple pendulum

A simple pendulum consists of an ideal string (i.e., weightless and inextensible) and a heavy point mass. The point mass is suspended through the string at a rigid support (also known as the point of suspension).

The point mass executes periodic and oscillatory motions. However, does it execute simple harmonic motion? Let us find out.

Suppose that a pendulum of length l is shifted by an angle θ from the equilibrium position, and then it is released. The trajectory of the point mass is shown by a yellow arc in the FBD of the point mass.

As the point mass is released, it starts oscillatory motion along the yellow arc due to the restoring torque provided by the gravitational force, mg. 

Two forces are acting on the point mass, which are: 

(i) Gravitational force, mg 

(ii) Tension (T) due to the string 

Therefore, the net torque on the point mass about point O is as follows: 

Since is not proportional to , the motion of the pendulum is not SHM. 

However, if we consider small oscillations, then

Where ( moment of inertia of point mass about O)

We know for angular SHM

 Therefore,

Hence, under small oscillation, the pendulum executes SHM with . on comparing

angular frequency,  

And time period,

And frequency, .

Note- Although the motion of a simple pendulum is always periodic and oscillatory, it executes SHM only when its angular displacement, θ, is small. For this reason, the formula of time period for the SHM executed by a simple pendulum, is valid only for small values of θ.

Seconds pendulum

A seconds pendulum is a simple pendulum whose time period is two seconds.It takes exactly one second for each swing (i.e., from one end to the other) in either direction.

Simple pendulum in accelerating frame

Suppose that there are two simple pendulums: one is in the ground frame (inertial frame) which is observed by observer O1, and the other is in an accelerating lift (non-inertial frame) which is observed by observer O2.

Thus, it is important to note that the motion of a simple pendulum in an accelerating lift to an observer in an inertial frame is neither periodic and oscillatory nor SHM.

To an observer O2 who is in the lift, the motion of the simple pendulum suspended in the lift will be SHM with the time period.

Where and is the acceleration of the non-inertial frame in which the pendulum is suspended.

As there is a pseudo force, , acting in the non-inertial frame opposite to the direction of the acceleration of the frame, g becomes geff in the formula of the time period of the simple pendulum suspended in a non-inertial frame. 

Now consider two case:

• Frame accelerating upwards: If the non-inertial frame moves with an upward acceleration a, then the pseudo acceleration will be in the downward direction; hence, the effective acceleration will be geff = (g + a). Vectorially, this can be written as follows:

Suppose that the non-inertial frame is moving up with acceleration,  

Therefore, the pseudo acceleration will be, . The pseudo force and gravitational force on the bob of the pendulum will be in the downward direction. 

Therefore, the effective acceleration will be, The time period of the oscillation will be,

Hence the time period decreases.

• Frame accelerating downwards: If the non-inertial frame moves with a downward acceleration a, then the pseudo acceleration will be in the upward direction; hence, the effective acceleration will be, geff = (g - a) 
   
• 

The time period of the oscillation will be,

Hence the time period increases.

Practice problem

Q.Two masses, MA and MB, are hung from two strings of lengths lA and lB, respectively. They are executing SHM with a frequency relation fA = 2fB, then which of the following equations is correct?

a. 
b. 
c.  and \\
d.  and

A.

The frequency of the simple pendulum is 

The frequency of the first mass is

The frequency of the first mass is

Also

Thus, option (A) is the correct answer.

Q. A pendulum is suspended from the ceiling of a truck accelerating uniformly on a horizontal road. What is the time period of the small oscillations about the mean position, If the acceleration is a0 and the length of the pendulum is l.

A. As soon as the truck starts moving with acceleration a0, the pendulum suspended from the ceiling of the truck becomes tilted from its mean position because of the pseudo force as shown in the figure.

Now, the pendulum will oscillate keeping the tilted position as the mean position of the oscillation. 

Therefore, the effective acceleration will be,

There for time period of the pendulum will be,

.

Q. A pendulum of 1 ft length suspended from the ceiling of an elevator accelerating upward has the time period . Find the acceleration of the elevator. (Take g = 32 ft s-2)

A. We have,

Length of the string of the simple pendulum, l = 1 ft 

Time period of oscillation, T =

Gravitational acceleration, g = 32 ft s-2

Suppose that the lift is moving with acceleration a. 

Therefore, the time period of the pendulum is,

Q.A simple pendulum is suspended from the roof of a trolley that moves in a horizontal direction with an acceleration a. Here, the time period is given by , what is the value of g′ equal to?

a. 
b. 
c. 
d. 

A. The FBD of the system is given as follows:

The net force on the bob is given as

Thus, option (D) is the correct answer.

FAQs

Q. Why does the pendulum stop after some time?
A. Pendulum stops because of air resistance.

Q. How does the length of the pendulum affect the time period?
A. If the length is more, the pendulum will take longer time to complete oscillations i.e greater time period and vise versa.

Q.What happens if some more weight is added to a simple pendulum.
A.As weight is added the bob will become heavier, But the effect of gravity will not change so it will swing at the same speed.

Q. What are the uses of a pendulum in real life?
A. Pendulum is used in metronomes, clocks, earth seismometers, and amusement parks rides etc.

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