Askiitians Tutor Team
Last Activity: 5 Months ago
To solve this problem, we need to analyze the forces acting on the rod when it is subjected to both the weight and the magnetic force. Let's break it down step by step.
Understanding the Forces at Play
Initially, the rod of mass m has a weight W acting downwards, given as 9 N. This weight causes an extension x₀ in the springs. When the battery is connected, a current i flows through the rod, which is placed in a magnetic field B. This setup generates a magnetic force F acting upwards on the rod.
Calculating the Forces
The maximum magnetic force acting on the rod is given as 4 N. Therefore, the total upward force when the current is flowing is the sum of the magnetic force and the force exerted by the springs due to their extension:
- Weight of the rod: W = 9 N
- Magnetic force: F = 4 N
Let’s denote the extension in the springs when the current is flowing as x. The total force exerted by the springs can be expressed using Hooke's Law:
F_s = K * x
Setting Up the Equation
When the current is flowing, the forces balance out as follows:
F_s + F = W
Substituting the known values into this equation gives:
K * x + 4 N = 9 N
From this, we can rearrange to find the force exerted by the springs:
K * x = 9 N - 4 N = 5 N
Finding the Extension
Now, we can express the extension x in terms of the spring constant K:
x = 5 N / K
Calculating the Fractional Extension
The fractional extension in the spring can be calculated by comparing the new extension x with the original extension x₀. The fractional extension is given by:
Fractional Extension = (x - x₀) / x₀
Since we know that x₀ is the initial extension caused by the weight of the rod alone, we can express it as:
K * x₀ = W = 9 N
Thus, we can find x₀:
x₀ = 9 N / K
Putting It All Together
Now we can substitute both extensions into the fractional extension formula:
Fractional Extension = (5 N / K - 9 N / K) / (9 N / K)
This simplifies to:
Fractional Extension = (5 N - 9 N) / 9 N = -4 N / 9 N = -4/9
However, since we are looking for the absolute value of the fractional extension, we can express it as:
Fractional Extension = 4/9
Final Thoughts
The fractional extension in the spring, when the magnetic force is applied, is 4/9. This means that the extension due to the magnetic force is significant compared to the original extension caused by the weight of the rod. Understanding how forces interact in this scenario is crucial for grasping concepts in mechanics and electromagnetism.
