A uniform disk with mass 8.5 kg and radius 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 8.1 revolution? The unit of the tangential velocity is m/s.
To find the magnitude v of the tangential velocity of a point on the rim of the disk after it has turned through 8.1 revolutions, we can use the principles of rotational motion.
Let's break down the problem step-by-step:
Step 1: Convert revolutions to radians
Given that the disk has turned through 8.1 revolutions, we need to convert this to radians. Since one revolution is equal to 2π radians, we can calculate:
θ = 8.1 revolutions × 2π radians/revolution
Step 2: Calculate the torque
The torque (τ) acting on the disk can be calculated using the equation:
τ = Frsin(θ)
Where F is the force applied to the rim, r is the radius of the disk, and θ is the angle between the force and the tangent to the rim.
Plugging in the values, we have:
τ = (31.5 N) × (8 m) × sin(35°)
Step 3: Calculate the moment of inertia
The moment of inertia (I) for a uniform disk rotating about its center can be calculated using the equation:
I = (1/2) × m × r^2
Where m is the mass of the disk and r is the radius of the disk.
Plugging in the values, we have:
I = (1/2) × (8.5 kg) × (8 m)^2
Step 4: Calculate the angular acceleration
The angular acceleration (α) can be calculated using the equation:
τ = I × α
Rearranging the equation, we get:
α = τ / I
Plugging in the values, we have:
α = (torque calculated in Step 2) / (moment of inertia calculated in Step 3)
Step 5: Calculate the final angular displacement
The final angular displacement (Δθ) can be calculated using the equation:
Δθ = α × t
Since the disk starts from rest, we can assume that the initial angular velocity is zero. Thus, we can use the equation to simplify:
v = α × t
But we already have Δθ from Step 1, so we can rewrite the equation as:
v = α × Δθ
Step 6: Calculate the tangential velocity
The tangential velocity (v) can be calculated using the equation:
v = ω × r
Where ω is the angular velocity and r is the radius of the disk.
Since the disk starts from rest, we can assume that the initial angular velocity is zero. Thus, we can use the simplified equation:
v = α × t × r
Plugging in the values, we have:
v = (α calculated in Step 4) × (Δθ calculated in Step 1) × (radius of the disk)
Simplify the equation by plugging in the given values, and calculate the final result.
To find the tangential velocity of a point on the rim of the disk, we can use the concept of torque and rotational dynamics.
Here's how you can solve this problem step by step:
Step 1: Convert the given angle from degrees to radians.
- Recall that the conversion factor from degrees to radians is π/180.
- In this case, the angle is 35 degrees, so the equivalent angle in radians can be calculated as (35 * π/180).
Step 2: Calculate the torque exerted on the disk.
- Torque (τ) is equal to the force applied (F) multiplied by the lever arm (r), where r is the radius of the disk.
- Given the force as 31.5 N and radius as 8 m, the torque can be calculated as (31.5 * 8).
Step 3: Determine the angular acceleration.
- The torque applied causes the disk to accelerate rotationally.
- The relation between torque and angular acceleration (α) is given by τ = I * α, where I is the moment of inertia of the disk.
- For a uniform disk rotating about its center, the moment of inertia is given by (1/2) * m * r^2, where m is the mass of the disk and r is the radius.
- Substituting the given values, the moment of inertia (I) can be calculated.
Step 4: Find the final angular displacement.
- The number of revolutions can be converted to an angular displacement in radians by multiplying 8.1 revolutions by 2π.
- This is because 1 revolution is equal to 2π radians.
Step 5: Use the rotational kinematic equation to find the final tangential velocity.
- The rotational kinematic equation relates the final tangential velocity (v) with the initial angular velocity (ω_0), angular acceleration (α), and angular displacement (θ).
- The equation is v^2 = ω_0^2 + 2αθ.
- Since the disk is initially at rest, ω_0 is equal to 0.
- Substitute the values of α, θ, and simplify the equation to find v.
Step 6: Take the square root of the calculated value to obtain v.
- The square root of the obtained value will give the magnitude of the tangential velocity.
By following these steps, you can calculate the magnitude v of the tangential velocity of a point on the rim of the disk.