A disk is free to rotate around its fixed axis of revolution (delta). The moment of inertia of disk around delta is J=4×10^-2 kg.m^2 This disk is initially at rest is submitted to the action of a couple whose moment is equal to 0.2 N.m What's the number of revolutions the disk performs during the first 2 seconds of motion
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To determine the number of revolutions the disk takes during the first 2 seconds of motion, we need to calculate the angular acceleration of the disk and then use it to find the angular displacement.
First, let's determine the angular acceleration (α) of the disk. We know that the moment of inertia (J) is equal to the product of the mass of the disk and the square of the radius (r) of rotation:
J = m × r^2
Since the disk is free to rotate around its fixed axis of revolution, the mass cancels out, giving us:
J = r^2
Now, substitute the moment of inertia value (J = 4×10^-2 kg.m^2) to find the radius (r) of rotation:
r^2 = 4×10^-2 kg.m^2
Solving for r, we get:
r ≈ √(4×10^-2 kg.m^2)
r ≈ 0.2 m
Next, let's calculate the torque (τ) applied to the disk using the equation:
τ = I × α
Where I is the moment of inertia and α is the angular acceleration. Since τ = 0.2 N.m and I = 4×10^-2 kg.m^2, we can rearrange the equation to solve for α:
α = τ / I
α = 0.2 N.m / (4×10^-2 kg.m^2)
α = 5 rad/s^2
Now, we can calculate the angular displacement (θ) of the disk during the first 2 seconds of motion using the equation:
θ = 0.5 × α × t^2
Where α is the angular acceleration and t is the time. Substituting the values, we get:
θ = 0.5 × 5 rad/s^2 × (2 s)^2
θ = 0.5 × 5 rad/s^2 × 4 s^2
θ = 10 rad
Since one revolution is equal to 2π radians, we can divide the angular displacement by 2π to find the number of revolutions:
Number of revolutions = θ / 2π
Number of revolutions ≈ 10 rad / (2π rad/rev)
Number of revolutions ≈ 1.59 revolutions
Therefore, during the first 2 seconds of motion, the disk will perform approximately 1.59 revolutions.