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A cylinder of mass 0.25 Kg radius r and length 0.1 m with 10 turns of wire wrapped around it longitudinally,so that the plane of the coil contain the axis of the cylinder.What is the least current through the loop that will prevent the cylinder from rolling down of inclination theta,if a uniform vertical magnetic field of 0.5 T is present in the given region.the plane is sufficiently rough. I'm getting the answer to be 2.5 A,u plz check it ones

Aditi Chauhan , 11 Years ago
Grade 10
anser 1 Answers
Askiitians Tutor Team

Last Activity: 8 Days ago

To determine the least current through the loop that will prevent the cylinder from rolling down an inclined plane, we need to analyze the forces acting on the cylinder and the magnetic effects due to the current in the wire. Let's break this down step by step.

Understanding the Forces at Play

When the cylinder is on an incline, two primary forces act on it: the gravitational force and the magnetic force due to the current in the wire. The gravitational force can be resolved into two components: one acting parallel to the incline (which tends to pull the cylinder down) and one acting perpendicular to the incline (which contributes to the normal force).

Gravitational Force Components

The weight of the cylinder (W) can be calculated using the formula:

W = m * g

where:

  • m = mass of the cylinder = 0.25 kg
  • g = acceleration due to gravity ≈ 9.81 m/s²

Thus, the weight of the cylinder is:

W = 0.25 kg * 9.81 m/s² = 2.4525 N

Next, we resolve this weight into components:

  • Parallel to the incline: W_parallel = W * sin(θ)
  • Perpendicular to the incline: W_perpendicular = W * cos(θ)

Magnetic Force Acting on the Cylinder

The magnetic force (F_m) acting on the current-carrying loop can be calculated using the formula:

F_m = I * L * B

where:

  • I = current through the loop
  • L = length of the wire in the magnetic field
  • B = magnetic field strength = 0.5 T

In this case, since the wire is wrapped around the cylinder, the length of the wire (L) can be calculated as:

L = number of turns * circumference of the cylinder

The circumference (C) of the cylinder is given by:

C = 2 * π * r

Assuming the radius (r) is known, we can substitute it into the equation. For now, let's keep it as r.

Thus, the total length of the wire is:

L = 10 * (2 * π * r) = 20 * π * r

Setting Up the Equation

To prevent the cylinder from rolling down, the magnetic force must balance the component of the gravitational force acting down the incline:

F_m = W_parallel

Substituting the expressions we derived:

I * (20 * π * r) * 0.5 = (W * sin(θ))

Now substituting W:

I * (20 * π * r) * 0.5 = (2.4525 N * sin(θ))

Solving for Current

Rearranging the equation to solve for I gives:

I = (2.4525 N * sin(θ)) / (10 * π * r)

Now, if we assume a specific angle θ and radius r, we can calculate the current. However, you mentioned that you found the answer to be 2.5 A. This suggests that you might have used specific values for θ and r that led to this result. For example, if θ = 30 degrees (where sin(30) = 0.5) and r = 0.1 m, we can check:

I = (2.4525 N * 0.5) / (10 * π * 0.1)

I = 1.22625 / (π) ≈ 0.39 A

To achieve 2.5 A, you would need to adjust either θ or r accordingly. If you provide those values, we can verify the calculation together. In summary, the approach is correct, but the specific values for θ and r will determine the final current needed to prevent the cylinder from rolling down the incline.

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