IEMDaily - Video Lecture Notes

Definition

Define Time period

The time taken by the wave for one complete oscillation for the density or pressure of the medium is called the time period. It is measured in seconds.

e.g. The time period of the minute hand is 60 minutes.

Diagram

Interpret length vs time period graphs of a simple pendulum

Time period of Pendulum

T = 2 π \sqrt{\frac{L}{g}}

hence,

T \propto L^{\frac{1}{2}}

The graph between time period and length becomes parabola as iterpreted in figure.

Diagram

Interpret length vs time period graphs of a simple pendulum

The following graph shows variation of time period with the length of pendulum.

Definition

Factors affecting the time period of a simple pendulum

T = 2 π \sqrt{\frac{l}{g}}

where

T :

Time period,

l :

length,

g :

acceleration due to gravity
1. The time period of oscillation is directly proportional to to the square root of its effective length.
2. The time period of oscillation is inversely proportional to the square root of acceleration due to gravity.
3. The time period of oscillation does not depend on the mass or material of the body suspended.
4. The time period of oscillation does not depend on the extent of swing on either side.

Example

Find frequency and time period of a simple pendulum

Example: A simple pendulum hangs from the ceiling of a car. Time period of pendulum when the car is at rest was found

2 s e c

. Now if the car moves uniformly in a large circle, pendulum stays in equilibrium with respect to car at an angle of

36^{0}

with the vertical. Find the new time period of the pendulum for small oscillations ? (use

c o s 36^{0} = 0.81

)

Solution:
We know that time period of simple pendulum is,

T = 2 π \sqrt{\frac{l}{g}} = 2 s e c - - - - - - - - - - (i)

Now it makes angle

36^{0}

when car is moving in a circle. Therefore
component of acceleration due to net force acting on the string will be

= \sqrt{g^{2} + a^{2}}

New Time period,

T^{'}

, will be

T^{'} = 2 π \sqrt{\frac{l}{\sqrt{g^{2} + a^{2}}}}

T^{'} = 2 π \sqrt{\frac{l}{g \sqrt{1 + \frac{a^{2}}{g^{2}}}}} - - - - - - - - - - - (i i)

Also,

t a n θ = a / g

--------

(i i i)

Solving

i, i i

and

i i i

, we get

T^{'} = 2 \sqrt{c o s θ}

= 2 \sqrt{0.81} = 1.8 s e c

Formula

Time period for a physical pendulum

Time period of Physical pendulum:

T = 2 π \sqrt{I / m g L_{c}}

where

L_{c}

is length of pendulum in cm.

Example

Period of a pendululm of infinite length

Time period of a pendulum of infinite length =

2 π \sqrt{l / g}

Where

l

tends to infinity.

Definition

Vernier Constant

The Vernier Constant is equal to the difference between values of one main scale division and one vernier scale division.
Vernier Constant: Value of 1 main scale division - Value of 1 vernier scale division

Definition

Zero error and its types

The zero error is equal to the length between the zero mark of the main scale and the zero mark of the vernier scale. (Here it's defined with respect to Vernier callipers)
Kinds of Zero Error:
(1) Positive Zero Error
(2) Negative Zero Error

Definition

Calculating zero error in vernier caliper

The attached diagram shows cases of zero error in a vernier caliper.
Case (a): No zero-error
Case (b): Positive zero-error of 3 vernier scale division (3rd line coinciding). Positive zero-error correction is done by subtracting the positive zero-error from the actual reading.
Case (c): Negative zero-error of 2 vernier scale division (8th line coinciding). Negative zero-error correction is done by adding the negative zero-error from the actual reading.