Askiitians Tutor Team
Last Activity: 5 Months ago
To find the tension in the string of a simple pendulum at various points in its motion, we need to consider the forces acting on the bob. The tension in the string varies depending on the position of the bob and its speed. Let's break down each part of your question step by step.
1. Tension at the Lowest Point
At the lowest point of the pendulum's swing, the forces acting on the bob include the gravitational force and the tension in the string. The gravitational force can be calculated using the formula:
- Weight (W) = mass (m) × gravitational acceleration (g)
Given:
- Mass (m) = 100 g = 0.1 kg
- Gravitational acceleration (g) = 9.81 m/s²
- Speed (v) at the lowest point = 1.4 m/s
First, calculate the weight:
W = 0.1 kg × 9.81 m/s² = 0.981 N
Next, we need to find the centripetal force required to keep the bob moving in a circular path. This force is provided by the tension in the string minus the weight of the bob:
- Centripetal Force (Fc) = m × v² / L
Where L is the length of the pendulum (1 m). Now, calculate the centripetal force:
Fc = 0.1 kg × (1.4 m/s)² / 1 m = 0.1 kg × 1.96 m²/s² = 0.196 N
Now, apply Newton's second law in the vertical direction:
T - W = Fc
T - 0.981 N = 0.196 N
T = 0.196 N + 0.981 N = 1.177 N
Rounding this gives us approximately:
T ≈ 1.2 N
2. Tension at an Angle of 0.2 Radians
When the pendulum bob is at an angle, the analysis changes slightly. The forces acting on the bob still include tension and weight, but we must consider the components of these forces. The gravitational force can be resolved into two components: one along the string and one perpendicular to it.
At an angle θ (0.2 radians), the components are:
- Weight component along the string: Wcos(θ)
- Weight component perpendicular to the string: Wsin(θ)
Using the same weight calculated earlier (0.981 N), we find:
Wcos(0.2) = 0.981 N × cos(0.2) ≈ 0.981 N × 0.9801 ≈ 0.962 N
Wsin(0.2) = 0.981 N × sin(0.2) ≈ 0.981 N × 0.1987 ≈ 0.195 N
The centripetal force is still given by:
Fc = m × v² / L = 0.196 N (as calculated previously).
Now, applying the forces in the radial direction:
T - Wcos(θ) = Fc
T - 0.962 N = 0.196 N
T = 0.196 N + 0.962 N = 1.158 N
Rounding this gives us approximately:
T ≈ 1.16 N
3. Tension at the Extreme Position
At the extreme position of the pendulum, the bob momentarily comes to rest before reversing direction. At this point, the only forces acting on the bob are the tension in the string and the weight. Since the bob is not moving, there is no centripetal force required.
At the extreme position, the tension in the string must balance the weight of the bob:
T = W
T = 0.981 N
Thus, the tension in the string when the bob is at the extreme position is:
T = 0.981 N
In summary, we calculated the tension in the string at various points of the pendulum's motion by analyzing the forces acting on the bob and applying Newton's laws. Each scenario required a different approach based on the position and speed of the bob.
