To solve this problem, we need to analyze the motion of a simple pendulum and how it behaves when released from a certain angle. The key here is to determine the distance of the nail from the point of suspension such that the pendulum bob will perform circular motion around the nail as the center. Let's break it down step by step.
Understanding the Pendulum's Motion
A simple pendulum consists of a mass (the bob) attached to a string of length L, which swings back and forth under the influence of gravity. When the pendulum is released from an angle, it converts potential energy into kinetic energy as it swings downwards.
Initial Conditions
In this scenario, the pendulum is released from an angle of 60 degrees with the vertical. The length of the pendulum is given as 1 meter. We need to find the vertical distance of the nail from the point of suspension that allows the bob to perform circular motion around the nail.
Finding the Height of the Nail
When the pendulum is released, it will swing downwards and reach its lowest point, where all potential energy is converted into kinetic energy. To perform circular motion around the nail, the bob must have enough speed at the lowest point to maintain tension in the string. The critical point occurs when the bob reaches the vertical position directly below the point of suspension.
- The vertical height of the bob from the point of suspension when it is at the lowest point is 0 meters.
- When the pendulum is at an angle of 60 degrees, we can find the vertical height of the bob above the lowest point using trigonometry.
Calculating the Vertical Distance
Using the cosine function, we can find the vertical height of the bob when released from 60 degrees:
Height above the lowest point (h) = L - L * cos(θ)
Substituting the values:
h = 1 - 1 * cos(60°) = 1 - 1 * (0.5) = 1 - 0.5 = 0.5 meters (or 50 cm)
Position of the Nail
For the bob to perform circular motion around the nail, the nail must be positioned at a distance equal to the radius of the circular path the bob will take. The radius of this circular path is equal to the length of the pendulum minus the height of the bob above the lowest point:
Radius = L - h = 1 - 0.5 = 0.5 meters (or 50 cm)
Now, since the bob will be moving in a circular path around the nail, the nail must be positioned below the point of suspension. Therefore, the distance of the nail from the point of suspension should be:
Distance of the nail = 50 cm below the point of suspension.
Final Answer
Given the options provided, the closest answer is:
- a) 80 cm above point of suspension
- b) 80 cm below point of suspension
- c) 60 cm above point of suspension
- d) 60 cm below point of suspension
None of the options directly match our calculated distance of 50 cm below the point of suspension. However, if we consider the closest option, we can conclude that the nail should be positioned below the point of suspension, but the exact distance is not listed among the choices.
