A straight conductor between points O and P has a mass of 0.54 kg and a length of 1.5 m and carries a current of 12 A (see figure below). It is hinged at O and is placed in a plane perpendicular to a magnetic field of 3.4 T. If the conductor begins from rest, determine the angular acceleration of the conductor due to the magnetic force. Ignore the effect of the gravitational force on the conductor. (Enter the magnitude.)
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To determine the angular acceleration of the conductor due to the magnetic force, we need to use the equation for the torque produced by a magnetic field on a current-carrying conductor.
The torque (τ) produced by the magnetic force can be calculated using the formula:
τ = μ × B × I × sin(θ)
Where:
- μ is the magnetic moment, equal to the product of the length (L) and mass (M) of the conductor: μ = M × L
- B is the magnetic field strength
- I is the current flowing through the conductor
- θ is the angle between the magnetic field and the direction of the current
However, since the conductor is hinged at point O and can rotate around it, the torque (τ) can also be expressed as:
τ = I × B × L × sin(θ) × d
Where:
- d is the distance from the point O to the center of mass of the conductor
In the given problem, the conductor is placed perpendicular to the magnetic field (θ = 90°), and we need to find the angular acceleration (α).
Since the conductor starts from rest, we can relate the torque (τ) to the moment of inertia (I) and the angular acceleration (α) using the equation:
τ = I × α
Therefore, the torque produced by the magnetic force is equal to the moment of inertia multiplied by the angular acceleration.
Now let's substitute the given values into the equation and solve for α.
Given:
Mass (M) = 0.54 kg
Length (L) = 1.5 m
Current (I) = 12 A
Magnetic field strength (B) = 3.4 T
First, calculate the magnetic moment (μ):
μ = M × L
μ = 0.54 kg × 1.5 m = 0.81 kg⋅m
Next, calculate the torque (τ):
τ = I × B × L × sin(θ) × d
Since θ = 90°, sin(θ) = 1
d = L/2 (assuming the center of mass is at the midpoint)
τ = I × B × L × sin(θ) × d
τ = (12 A) × (3.4 T) × (1.5 m) × (1) × (1.5 m/2)
τ = 183.6 N⋅m
Finally, calculate the angular acceleration (α):
τ = I × α
α = τ / I
α = 183.6 N⋅m / (0.54 kg × 1.5 m^2)
α ≈ 229.45 rad/s^2
Therefore, the approximate angular acceleration of the conductor due to the magnetic force is 229.45 rad/s^2.