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Fill in the Blanks. \(\frac{\mathrm{T}}{2}\) The time taken to perform one to-and-fro motion (or) from one extreme position to other extreme position and back is called time period (T).

Short Answer

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Question: In a to-and-fro motion, the time taken to perform half a cycle, such as moving from one extreme position to the other extreme position, is represented by the fraction ______. Answer: \(\frac{\mathrm{T}}{2}\)

Step by step solution

01

Understand Time Period

The time period (T) for a to-and-fro motion represents the amount of time it takes for an object (such as a pendulum) to make one complete cycle, starting and ending at the same point in its journey. This includes moving away from the starting point to its extreme position, then back towards and passing the starting point, until it reaches the other extreme position, and then returning to the starting point again.
02

Relate to the given fraction

Now that the concept of the time period is understood, let's focus on the given fraction \(\frac{\mathrm{T}}{2}\). This represents half of the time period (T). It means that this amount of time corresponds to half a cycle of the motion.
03

Fill in the Blank

As the exercise asks to fill in the blank to understand the given fraction, we can mention that \(\frac{\mathrm{T}}{2}\) corresponds to the time taken to perform half of a to-and-fro motion, i.e., moving from one extreme position to the other extreme position (or) from the starting point to the other extreme position without returning.
04

Summary

In summary, the given fraction \(\frac{\mathrm{T}}{2}\) denotes the time taken to perform half a to-and-fro motion, which can be understood as either moving from one extreme position to the other extreme position or moving from the starting point to the other extreme position without returning. This understanding helps to fill in the blank and ensures a clear idea of how the fraction relates to the time period (T) of a to-and-fro motion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

To-and-Fro Motion
To-and-fro motion is a fundamental concept in understanding movements like those of a pendulum. Imagine you are swinging on a playground swing. When you go forward to the highest point you can reach, and then come back to the point where you started, that's an example of to-and-fro motion.
In scientific terms, this type of motion is also known as periodic motion, as it repeats after a certain interval of time. It's essential to grasp this concept to understand phenomena in physics, such as oscillations and waves.
Understanding to-and-fro motion can help you see how many systems in nature and human-made environments work, like the ticking of a clock or the gentle rocking of a boat.
  • It involves movement moving back and forth.
  • It's continuous and repeats after a time period.
  • Common examples include pendulum swings and heartbeats.
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. This is one of the simplest examples of a system that uses to-and-fro motion.
When you pull a pendulum to one side and let it go, it swings in an arc to one extreme position, then back to the original position, and continues to the opposite extreme position. Eventually, it returns to the starting point, completing its cycle.
The movement of a pendulum is widely used to illustrate the concepts of time period and oscillation. It's interesting to note how pendulums have been utilized in clocks for centuries to keep time.
  • Pendulums demonstrate consistent periodic motion.
  • Used in timekeeping and scientific experiments.
  • Swinging through its path displays principles of gravity and inertia.
Oscillation
Oscillation refers to the repeated back and forth motion around a central value or position, akin to a pendulum's swings from side to side.
It plays a critical role in various fields such as physics, engineering, and even biology. Oscillations can be found in vibrating guitar strings, the alternating current in electrical circuits, and the cycles of planetary orbits.
In the context of a pendulum, each full cycle of motion—moving to one side, returning, and moving to the other side before coming back—is considered one complete oscillation.
  • Oscillations help describe wave-like phenomena.
  • They occur in both mechanical systems and electromagnetic fields.
  • Understanding oscillation is key to fields like communications and acoustics.
Extreme Position
The extreme position in to-and-fro motion or oscillations is the point where the object reaches its maximum displacement from the central position before reversing direction.
In the case of a pendulum, these extreme positions are the farthest points the pendulum bob rises during its swing. It's akin to the highest points where a playground swing reaches before it changes direction and comes back down.
Recognizing these points is crucial because the time it takes to move from one extreme position to the other and back defines one complete cycle of the motion, thereby determining the time period.
  • Extreme positions mark the limits of an object's movement.
  • They are essential in calculating the time period of motion.
  • These points relate to energy exchange in oscillatory systems.

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Most popular questions from this chapter

Motion along a straight line. (a) The motion of coin on a carrom board. (b) The motion of air bus in straight line. Circular motion \(\quad\) (a) Earth revolving around the sun. (b) Potter's wheel. Periodic motion \(\quad\) (a) The swinging pendulum of a wall clock. (b) Halley commet visits earth at regular periods.

Fill in the Blanks. vibratory The molecules in solid undergo vibratory motion.

The average distance per unit time, when the body is moving with variable speed, is called average speed, Average speed \(=\frac{\text { Total distance travelled }}{\text { Total time taken }}\)

The different kinds of motion are: (i) Translatory motion: \(\mathrm{A}\) bus moving on a road, the motion of a rising balloon, the free fall of a stone under gravity, the motion of a cricket ball when it is hit by a batsman are examples of translatory motion. Translatory motion is further classified as rectilinear motion and curvilinear motion. When an object moves along a straight path, its motion is said to be rectilinear motion. The marching of soldiers on a straight road, the motion of a car on a straight road, the motion of carrom board coin are examples of rectilinear motion. When an object moves along curved path, its motion is called curvilinear motion. A bus moving on a fly-over bridge, a car taking a turn, a football kicked from the ground into air all have curvilinear motion. (ii) Rotatory motion: In this type of motion, the object rotates about a fixed axis. The motion of blades of a ceiling fan, the spin motion of a top, the motion of turbine, the motion of the earth around the sun are all circular or rotatory motion. In some cases, the rotatory and translatory motions take place simultaneously. When a bicycle moves, its wheels undergo translatory and rotatory motion. (iii) Oscillatory motion: A boy on a swing moves to-and-fro (back and forth). The motion such as above, where an object moves to-and-fro is called oscillatory motion. Other examples of oscillatory motion are the motion of the pendulum of a clock, the motion of a needle of a sewing machine, the motion of a piston of an engine etc.

Take a metre scale and measure the length of the string from the point of suspension to the lower tip of the bob \(\left(\ell_{1}\right)\) (b). Now, place the bob over a meter scale and hold it in position with two wooden blocks or stiff cardboards and measure the diameter (D) of the bob (c). Calculate the radius \(\mathrm{R}\) of the bob by dividing diameter by 2 (a). Then the length of the pendulum \(\ell=\left(\ell_{1}-\mathrm{R}\right)(\mathrm{d})\). Consider the formula \(\mathrm{T}\) \(=2 \pi \sqrt{\frac{\ell}{\mathrm{g}}}\) and find the time period of the simple pendulum by substituting the value of \({ }^{\prime} \ell^{\prime}(\mathrm{e})\).

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