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.
