Question

A simple pendulum is executing S.H.M. around point $\mathrm{O}$ between the end points $B$ and $C$ with a periodic time of $6\mathrm{s}$. If the distance between $\mathrm{B}$ and $\mathrm{C}$ is $20\mathrm{cm}$ then in what time will the bob move from $C$ to $D$ ? Point $D$ is at the mid-point of $C$ and $\mathrm{O}$. (A) $1\mathrm{s}$ (B) $2\mathrm{s}$ (C) $3\mathrm{s}$ (D) $4\mathrm{s}$

Short answer

The time it takes for the pendulum bob to move from point C to point D is $3\mathrm{s}$.

Step-by-step solution

Determine the total travel distance and distance from C to D

For the pendulum performing S.H.M., the total distance from point B to point C is 20 cm. To find the distance between point C and point D, we must determine the mid-point between C and O. The distance between C and D is half the distance between C and O. Since the distance between B and C is 20 cm, we can say that the distance between O and B, or O and C is half of this value, which is 10 cm. Therefore, the distance between point C and point D is $$CD=\frac{1}{2}(CO)=\frac{1}{2}(10cm)=5cm$$

Determine the angular displacement in radians from C to D

In this step, we need to find angular displacement from C to D. Since it's a pendulum, so we can use the formula $$\theta ={\sin }^{-1}\frac{x}{L}$$ where $\theta$ is the angular displacement, $x$ is the linear displacement and $L$ is the length of the pendulum from point O to the either B or C. Here, we have $x=CD=5cm$. We need to find $L$, the length of the pendulum, which can be calculated by dividing $BC$ by $2\sin \theta$. Since $BC=20cm$, $$L=\frac{BC}{2\sin \theta }=\frac{20\mathrm{cm}}{2\sin \theta }=\frac{10\mathrm{cm}}{\sin \theta }$$ Now, plugging L and x in the formula, we get $$\theta ={\sin }^{-1}\frac{5\mathrm{cm}}{\frac{10\mathrm{cm}}{\sin \theta }}$$ After simplification, we find $$\theta ={\sin }^{-1}\frac{1}{2}={\sin }^{-1}\left(\frac{1}{2}\right)={30}^{\circ }(\text{or}\frac{\pi }{6}\text{radians})$$ So, the angular displacement from C to D is $\frac{\pi }{6}$ radians.

Determine the time it takes to travel from C to D

Here, the simple pendulum is performing S.H.M., and it is mentioned that its periodic time is 6 seconds. The relation between the angular displacement and time for a simple pendulum is given by the formula: $$T(t)=\frac{2\pi \sqrt{\frac{L}{g}}}{\theta }$$ Where $T(t)$ is the time it takes for the object to move through the angle $\theta$, $L$ is the length of the pendulum, and $g$ is the acceleration due to gravity. Now, the periodic time for this pendulum is 6 seconds. From our previous calculation of angular displacement, the pendulum moves through an angle of $\frac{\pi }{3}$ radians between B and C. Therefore, we can use this information to determine the time it takes to go through $\frac{\pi }{6}$ radians (from C to D): $$T(t)=\frac{6}{2}=3\mathrm{s}$$

Check and choose the answer

After calculating the time it takes for the pendulum to travel from point C to point D, we get it as 3 seconds. So now, we can match our answer to the options given in the problem: (A) $1\mathrm{s}$ (B) $2\mathrm{s}$ (C) $3\mathrm{s}$ (D) $4\mathrm{s}$ The correct answer is (C) $3\mathrm{s}$.

Key concepts

Physics Education

Physics education often seeks to break down complex concepts into simpler, digestible parts, much like the step-by-step exploration of a pendulum's motion. Learning about simple harmonic motion (SHM) is essential in understanding how repetitive and periodic motion occurs in our world. In a physics classroom or through educational platforms, educators strive to make topics like pendulum motion approachable, piquing interest and fostering deep comprehension.

By focusing on practical examples such as the simple pendulum, students can see physics theories in action. The pendulum demonstrates key principles of SHM, a critical concept in physics education that illustrates periodic motion without energy loss under ideal conditions. Through hands-on experiments and guided problem-solving, physics education equips students with the necessary tools to analyze and solve real-world problems.

• Key concepts like time periods and displacement become less abstract, more tangible.
• Understanding SHM lays the groundwork for advanced studies in oscillations and waves.
• Effective physics education promotes critical thinking and problem-solving skills.

Pendulum Motion

Pendulum motion is a classic example of simple harmonic motion. In a simple pendulum, a weight (or bob) is suspended from a pivot point and swings back and forth in an arc under the influence of gravity. This motion is periodic, meaning it repeats at regular intervals.

The pendulum's path forms an arc between two extreme points, and its motion can be described by the concepts of amplitude, frequency, and period. The amplitude is the maximum angular displacement from the rest position. In the discussed exercise, the pendulum swings between points B and C, covering a total distance of 20 cm.

• The period is the time it takes for one full cycle of motion, which in this case is 6 seconds.
• The frequency is the number of cycles per second and is the inverse of the period.
• Pendulum motion is exemplified without energy loss in ideal conditions, showing pure SHM principles.

Understanding pendulum motion helps illustrate how mechanical systems respond to forces and energy, a principle found in multiple engineering and technological applications.

Angular Displacement

Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. In the context of a pendulum, it is essential for describing the pendulum's position relative to its rest, or equilibrium point.

In the exercise presented, angular displacement plays a crucial role in determining the motion as the pendulum swings between points C and D. The problem involves calculating how far the pendulum rotates when moving between these two points. This displacement was calculated as $\theta =\frac{\pi }{6}$, meaning the pendulum moves through an angle of 30°.

• The formula $\theta ={\sin }^{-1}\frac{x}{L}$ connects linear and angular displacement.
• Determining angular displacement involves understanding trigonometric relationships inherent in pendulum motion.
• Comprehension of angular displacement is vital in fields like robotics and mechanical systems.

Exploring this concept helps students transition from basic trigonometry to applications in circular dynamics and oscillations.

Periodic Time

Periodic time, or simply "period," is the duration taken by a pendulum to complete one full cycle of its motion. In the described exercise, the pendulum's periodic time is 6 seconds, meaning every 6 seconds, it completes one swing from point B to C and back.

This understanding of periodicity helps clarify the nature of oscillatory systems. By calculating how the pendulum divides its journey based on its periodic time, we can make inferences about time intervals for smaller motions such as moving from point C to D in 3 seconds.

• The period provides insights into the time-specific aspects of motion and helps solve related timing questions.
• Ample exposure to periodic concepts gives students the analytical tools needed for advanced physics topics, like wave mechanics.
• Analyzing periodicity enriches problem-solving techniques used across various science and engineering disciplines.

Understanding periodic time is pivotal in grasping the synchronization in systems ranging from simple pendulums to complex machinery.