Question

How does the time period of a swing change if a person sitting on it stands? What will happen if another person of same mass joins the first one?

Answer 1

Time period(T):

1. The time taken to complete one oscillation is called time period.
2. It is denoted by the letter T.
3. Its SI or CGS unit is second (s).
4. It is also called as the period of oscillation.

Factors on which time period of a simple pendulum depends:

1. Two factors on which the time period of a simple pendulum depends are : Length of the pendulum (l) and Acceleration due to gravity (g).
2. Time period of a simple pendulum is given by the formula,
   $$\begin{gathered} T=2𝞹\sqrt{\frac{l}{g}} \\ where,l=effectivelengthofpendulum \\ g=accelerationduetogravity \\ Valueofg=9.8m/s^2 \end{gathered}$$

The effective length(l) of the pendulum:

1. The distance between the point of suspension and the center of gravity of the bob is called effective length.
2. The period (T) of a simple pendulum is directly proportional to the square root of the effective length of the pendulum provided the amplitude is small.

$$\begin{gathered} T=\sqrt{l} \\ where,l=effectivelengthofpendulum \\ {T}^2\alpha l \end{gathered}$$

The time period of the swing when the person sitting on the swing stands:

1. The swinging of a person is considered a simple pendulum.
2. When the person stands up on the swing the distance between the point of suspension and the center of gravity(C.G.) decreases.
3. Thus effective length(l) is reduced.
4. As we know the period (T) of a simple pendulum is directly proportional to the square root of the effective length of the pendulum, and the time period of oscillation decreases.

where,

$$\begin{gathered} L=orginaleffectivelengthofapendulum \\ l=reducedeffectivelengthofapendulum \end{gathered}$$

Mass of the bob:

1. At a given time place,the time period of a simple pendulum does not depend on the mass of the bob provided the effective length remains the same and the amplitude is not large.

When another person of the same mass joins the first one.

1. When another person of the same mass joins the first one, the time period does not change
2. Because the time period is independent of the mass.
3. Thus time period will remain the same.

Hence, the time period of a swing decreases when a person sitting on a swing stands, and the time period will remain the same when another person of the same mass joins the first one.