Question

In the experiment of simple pendulum, we have taken a thread of \(140 \mathrm{~cm}\), and an amplitude of \(5 \mathrm{~cm}\) to begin with. Here \(\theta\) to begin with is about........ (A) \(5^{\circ}\) (B) \(8^{\circ}\) (C) \(2^{\circ}\) (D) \(3^{\circ}\)

Short answer

(C) \(2^\circ\)

Step-by-step solution

Understand the simple pendulum and the given parameters

A simple pendulum is an idealized model consisting of a weight (mass) attached to a string that is free to swing from a pivot point. The length of the pendulum is the distance between the pivot point and the center of mass of the weight, which in our case, is 140 cm. The amplitude of the pendulum refers to the maximum horizontal displacement of the weight from its equilibrium position. In our case, that is 5 cm.

Calculate the initial angle

We know that when the pendulum is at its maximum amplitude, the horizontal and vertical displacement together make a right-angled triangle with the length of the pendulum as one side (the hypotenuse). To find the angle θ, we'll use the sine function in trigonometry. In a right-angled triangle, sin(θ) = (opposite side) / (hypotenuse) or sin(θ) = (amplitude) / (length of the pendulum). Now, we are given amplitude = 5 cm and length = 140 cm. sin(θ) = (5 cm) / (140 cm) We need to find θ; we can use the inverse sine function (arcsin) to do this: θ = arcsin(5 cm / 140 cm)

Calculate the angle

θ = arcsin(5 cm /140 cm) θ ≈ arcsin(0.0357) (rounded to 4 decimal places) θ ≈ 2.0494° (rounded to 4 decimal places) Since we can't have a decimal degree value in the answer, we should round it to the nearest whole number, which gives: θ ≈ 2° So, the initial angle of the pendulum is approximately 2°. Therefore, the correct answer is: (C) \(2^\circ\)

Key concepts

Understanding Amplitude in a Simple Pendulum

In the context of a simple pendulum, understanding amplitude is fundamental. Amplitude refers to the maximum distance the pendulum bob moves from its resting position. It is essentially the maximum extent of vibration or oscillation measured from the position of equilibrium. In simpler terms, it's how far the pendulum swings to one side. In this experiment, the amplitude is specified as 5 cm. This value is significant because it indicates that the furthest point the pendulum reaches from its central position is 5 cm horizontally.
The amplitude affects how the pendulum will swing, and it's a key factor in determining other properties of the motion, such as the initial angle (θ). This can be seen when using trigonometry to find the angle. Given the importance of amplitude, it plays a crucial role in calculations involving pendulum motion.

Role of Trigonometry with the Simple Pendulum

Trigonometry comes into play significantly when dealing with a simple pendulum, especially for calculating angles like θ. When the pendulum oscillates, its path forms a right-angled triangle.
In this scenario, the length of the pendulum serves as the hypotenuse, and the horizontal displacement, also known as the amplitude, is the opposite side of the angle θ. The sine function in trigonometry relates these sides of the triangle.

• The sine of an angle in a right triangle is calculated as the length of the opposite side divided by the hypotenuse.
• This concept is used to express the relationship as: sin(θ) = (amplitude) / (length of the pendulum).

By using this trigonometric identity, one can solve for the angle of the pendulum. This makes trigonometry a powerful tool in analyzing pendulum motion and allows us to understand the position and the dynamics of the pendulum based on given lengths.

Using the Inverse Sine Function to Calculate Angles

The inverse sine function, also known as the arcsine function, is a critical tool for finding angles in the context of pendulum motion. Once you have the sine of an angle, you may need the actual angle measurement itself, which is where arcsin becomes useful.
For our simple pendulum problem, after calculating the sine of angle θ as the ratio of the amplitude to the pendulum's length (sin(θ) = 0.0357), you need to find θ which gives the angle in degrees or radians using arcsin.

• Arcsin is the function that "undoes" the sine, allowing us to retrieve the angle from its sine value.
• For example, if sin(θ) = 0.0357, you calculate θ = arcsin(0.0357).

The use of the inverse sine function is thus essential for converting between the trigonometric ratio and the actual angle. With our premise, this function yielded an angle of approximately 2°, showing its practical importance in real-world pendulum calculations.