Suppose the amplitude of a simple pendulum having a bob pf mass m is Q.Find the tension in the string when the bob is at its extreme position.

To determine the tension in the string of a simple pendulum when the bob is at its extreme position, we need to analyze the forces acting on the bob at that point. The extreme position is where the pendulum reaches its maximum displacement from the vertical, and at this point, the bob momentarily comes to rest before reversing direction.

Understanding Forces at the Extreme Position

At the extreme position, two main forces act on the bob:

• Gravitational Force (Weight): This force acts downward and is given by the equation \( F_g = mg \), where \( g \) is the acceleration due to gravity.
• Tension in the String (T): This force acts upward along the string, counteracting the weight of the bob.

Applying Newton's Second Law

At the extreme position, the bob is not moving vertically; hence, the net force acting on it is zero. This leads us to the equation:

\( T - mg = 0 \)

From this, we can deduce that:

\( T = mg \)

Considering the Amplitude

The amplitude \( Q \) of the pendulum does not directly affect the tension at the extreme position in terms of the formula we derived. However, it does influence the height to which the bob rises. The height \( h \) at the extreme position can be calculated using trigonometry, where:

\( h = L - L \cos(\theta) \)

Here, \( L \) is the length of the pendulum, and \( \theta \) is the angle corresponding to the amplitude \( Q \). The relationship between \( Q \) and \( \theta \) can be established through:

\( Q = L \sin(\theta) \)

Final Calculation of Tension

To find the tension in the string at the extreme position, we can summarize our findings:

1. The tension \( T \) is equal to the weight of the bob at that point:

\( T = mg \)

2. The amplitude \( Q \) helps us understand the maximum height but does not change the tension calculation directly.

Example Calculation

Let’s say the mass \( m \) of the bob is 2 kg. The acceleration due to gravity \( g \) is approximately \( 9.81 \, \text{m/s}^2 \). Plugging in these values:

\( T = 2 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 19.62 \, \text{N} \)

Thus, the tension in the string when the bob is at its extreme position is 19.62 N.

In summary, the tension in the string at the extreme position of a simple pendulum is equal to the weight of the bob, which remains constant regardless of the amplitude of the swing. This understanding is crucial for analyzing pendulum motion and the forces involved.