A simple pendulum was given to a physics student to determine its time period. Arrange the following steps in sequential order to determine its time period. (a) Calculate the radius of the bob ' \(R\) ' by dividing the diameter by 2 . (b) Take a metre scale and measure the length of the string from the point of suspension to the lower tip of the bob \((\ell)\). (c) Place the bob over a meter scale and hold it in position with two wooden blocks or stiff cardboards. Measure the diameter (D) of the bob. (d) Now, the length of the pendulum ( \(\ell\) ) is given by \(\left(\ell_{1}-\mathrm{R}\right)\). (e) Consider the formula \(\mathrm{T}=2 \pi . \sqrt{\frac{\ell}{g}}\) Calculate the time period of the simple pendulum. (1) abcde (2) edcba (3) bcade (4) deabc

Understanding the time period of a pendulum is fundamental in the study of oscillatory motion. It is the time taken for a pendulum to make one complete back-and-forth swing, and is usually denoted by the symbol 'T'. What's intriguing about the time period is that for small angles (less than about 15 degrees), it does not depend on the amplitude of the swing or the mass of the bob, but rather on the length of the pendulum and the acceleration due to gravity. Therefore, pendulums of different sizes with identical lengths and in the same gravitational field swing with the same period, a phenomenon which can be quite surprising the first time one encounters it. This characteristic makes pendulums useful for timekeeping.

To measure the time period practically, students often use a stopwatch to time several oscillations and then divide by the number of oscillations to reduce the error.

The length of a pendulum string is a crucial variable impacting its swinging motion. This length, usually symbolized by the Greek letter \(\ell\), is defined as the distance from the fixed point where the pendulum is suspended to the center of mass of the pendulum's bob. To accurately measure it, one must begin at the point of suspension and extend the measuring tape or meter stick straight down to the bottom tip of the bob.

Remember, it's not the entire string length but the distance to the pendulum's center of mass that affects its period. Keeping the string taut and straight while measuring is key. Furthermore, ensuring the precision of this measurement is vital, as even small errors can significantly affect the calculated time period of the pendulum.

In calculating the time period of a pendulum, another essential step involves determining the exact radius of the pendulum's bob. This measurement is critical since the length of the pendulum used in calculations is the distance from the point of suspension to the bob's center of mass, which requires knowing the bob's radius.

To calculate the radius, \(R\), one must first accurately measure the diameter of the bob by placing it against a ruler or caliper. Once the diameter is obtained, the radius is found by simply dividing the diameter by two. It's beneficial to ensure that the diameter is measured along the widest point of the bob, and that the measurement is as precise as possible to avoid adding further uncertainty to the final time period computation.

The formula \(T = 2\pi \sqrt{\frac{\ell}{g}}\) elegantly encapsulates the relationship between the physical properties of a simple pendulum and its time period. Here, 'T' represents the time period of the pendulum, '\(\ell\)' is the effective length of the pendulum, and 'g' is the acceleration due to gravity. The factor of \(2\pi\) stems from the properties of circular motion, as pendulum motion can be viewed as a projection of uniform circular motion.

The beauty of this formula lies in its simplicity and universally applicable nature for small-amplitude oscillations. It indicates that the time period is proportional to the square root of length and inversely proportional to the square root of the gravitational acceleration. Typically, 'g' is taken as \(9.81 m/s^2\) on the Earth's surface, though this value can vary slightly depending on location. The formula allows for easy prediction of how changes in the length of the pendulum will affect the time period, a fundamental principle in understanding harmonic motion.