A uniform disk with mass 6 kg and radius 1.8 m is pivoted at its center about a horizontal, frictionless axle that is stationary. The disk is initially at rest, and then a constant force 31.5N is applied to the rim of the disk. The force direction makes an angle of 35 degree with the tangent to the rim. What is the magnitude v of the tangential velocity of a point on the rim of the disk after the disk has turned through 5.9 revolution? The unit of the tangential velocity is m/s.
tau = I alpha
tau = Fcos35
I you'll have to look up
then lastly
omega^2 = 2 alpha θ and v = r omega
is £c 35 degrees???
i dont kno
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To find the magnitude of the tangential velocity of a point on the rim of the disk after it has turned through 5.9 revolutions, we can follow these steps:
1. Determine the angular acceleration (α) of the disk.
2. Calculate the time (t) taken for the disk to complete 5.9 revolutions.
3. Find the angular displacement (θ) of the disk using the number of revolutions.
4. Use the equation for angular motion to calculate the final angular velocity (ω) of the disk.
5. Convert the angular velocity (ω) to the tangential velocity (v) using the equation v = ω * r, where r is the radius of the disk.
Let's go through each step in detail:
Step 1: Determine the angular acceleration (α) of the disk.
The torque applied to the disk is given by the equation τ = I * α, where τ is the torque and I is the moment of inertia of the disk. Since the disk is rotating about its center, the moment of inertia is equal to 1/2 * m * r^2, where m is the mass of the disk and r is its radius.
τ = I * α becomes 31.5 * r * sin(35°) = (1/2) * m * r^2 * α (Note: τ = r * F * sin(θ))
Simplifying, we get α = (2 * 31.5 * sin(35°)) / (m * r)
Step 2: Calculate the time (t) taken for the disk to complete 5.9 revolutions.
One revolution is equal to 2π radians. Therefore, 5.9 revolutions is equal to 5.9 * 2π radians.
Using the equation relating the angular displacement (θ), the initial angular velocity (ω₀), the final angular velocity (ω), and time (t): θ = ω₀ * t + (1/2) * α * t², with ω₀ = 0 and θ = 5.9 * 2π, we can solve for t.
Step 3: Find the angular displacement (θ) of the disk using the number of revolutions.
θ = 5.9 * 2π
Step 4: Use the equation for angular motion to calculate the final angular velocity (ω) of the disk.
Using the equation ω = ω₀ + α * t, we can substitute ω₀ = 0 and the calculated value of α to solve for ω.
Step 5: Convert the angular velocity (ω) to the tangential velocity (v) using the equation v = ω * r.
Using the calculated value of ω and the given radius (r), we can find v.
By following these steps, we can find the magnitude of the tangential velocity (v) of a point on the rim of the disk after it has turned through 5.9 revolutions.