To solve this problem, we need to analyze the forces acting on the ball and the work done by Harry as he pulls the thread. Let's break it down step by step.
Understanding the Scenario
We have a ball with a mass of 0.1 kg tied to a thread. Initially, the ball is moving in a circular path with a radius of 0.4 m, and Harry is pulling the thread with a force of 0.8 N, keeping the radius constant. Later, he increases the force to 4 N, reducing the radius to 0.1 m. We need to calculate the work done by Harry and the work done by the ball during this process.
Work Done by Harry
The work done by a force is calculated using the formula:
- Work (W) = Force (F) × Distance (d) × cos(θ)
In this case, the force exerted by Harry is in the direction of the movement of the ball, so θ = 0 degrees, and cos(0) = 1. Therefore, the formula simplifies to:
Next, we need to find the distance the ball travels as the radius changes from 0.4 m to 0.1 m. The distance can be calculated as the difference in the circumference of the circles formed by the two radii:
- C1 = 2π × r1 = 2π × 0.4 m
- C2 = 2π × r2 = 2π × 0.1 m
Calculating these:
- C1 = 2π × 0.4 ≈ 2.51 m
- C2 = 2π × 0.1 ≈ 0.63 m
The distance the ball moves while the radius decreases is:
- d = C1 - C2 ≈ 2.51 m - 0.63 m ≈ 1.88 m
Now, substituting the values into the work formula:
- W = 4 N × 1.88 m ≈ 7.52 J
Work Done by the Ball
To find the work done by the ball, we need to consider the change in kinetic energy. Since the ball is moving in a circular path, we can use the formula for kinetic energy:
However, we need to determine the speed of the ball. The tension in the thread provides the centripetal force necessary for circular motion:
Initially, when the radius is 0.4 m, the tension is 0.8 N:
- 0.8 N = 0.1 kg × (v² / 0.4 m)
Solving for v²:
- v² = (0.8 N × 0.4 m) / 0.1 kg = 3.2 m²/s²
Now, calculating the initial kinetic energy:
- K.E.1 = 0.5 × 0.1 kg × 3.2 m²/s² = 0.16 J
Next, when the radius is reduced to 0.1 m, the tension is now 4 N:
- 4 N = 0.1 kg × (v² / 0.1 m)
Solving for v² again:
- v² = (4 N × 0.1 m) / 0.1 kg = 4 m²/s²
Calculating the final kinetic energy:
- K.E.2 = 0.5 × 0.1 kg × 4 m²/s² = 0.2 J
The work done by the ball is the change in kinetic energy:
- Work done by the ball = K.E.2 - K.E.1 = 0.2 J - 0.16 J = 0.04 J
Final Results
To summarize:
- Work done by Harry = 7.52 J
- Work done by the ball = 0.04 J
This analysis shows how forces and motion interact in a circular path, and how work is calculated based on the changes in energy and distance. If you have any further questions or need clarification on any part, feel free to ask!
