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) \(2^\circ\)