A disk with mass m = 8.1 kg and radius R = 0.33 m begins at rest and accelerates uniformly for t = 18 s, to a final angular speed of ω = 34 rad/s.
What is the angular acceleration of the disk?
What is the angular displacement over the 18 s?
What is the moment of inertia of the disk?
What is the change in rotational energy of the disk?
What is the tangential component of the acceleration of a point on the rim of the disk when the disk has accelerated to half its final angular speed?
What is the magnitude of the radial component of the acceleration of a point on the rim of the disk when the disk has accelerated to half its final angular speed?
What is the final speed of a point on the disk half-way between the center of the disk and the rim?
What is the total distance a point on the rim of the disk travels during the 18 seconds?
calculate moment of inertia I
alpha = change in angular rate/change in time
= (34-0)/18 radians/second^2
Theta = (1/2) alpha t^2
Ke = (1/2) I w^2
tangent acceleration at rim = r * alpha
radial acceleration = v^2/r = w^2 r
where w = (1/2) 34
w (r/2)
d = average speed * time
but average speed = 1/2 final speed
d = (1/2) r w * t
d = (1/2) .33 (34) (18)
To answer these questions, we'll need to use some basic rotational motion equations. Let's go through each question step by step:
1. Angular acceleration (α):
Angular acceleration is the rate at which the angular velocity changes. We can calculate it using the formula:
α = (ω - ω₀) / t
where ω₀ is the initial angular velocity (which is zero in this case), ω is the final angular velocity, and t is the time taken.
In this case, ω = 34 rad/s and t = 18 s, so we can substitute these values into the formula to find α.
2. Angular displacement (θ):
Angular displacement is the angle through which an object rotates. We can calculate it using the formula:
θ = ω₀ * t + (1/2) * α * t²
where ω₀ is the initial angular velocity, t is the time taken, and α is the angular acceleration.
In this case, ω₀ = 0 rad/s, α can be obtained from the previous step, and t = 18 s, so we can substitute these values into the formula to find θ.
3. Moment of inertia (I):
The moment of inertia measures an object's resistance to changes in rotational motion. For a solid disk, the moment of inertia formula is:
I = (1/2) * m * R²
where m is the mass of the disk and R is the radius.
In this case, m = 8.1 kg and R = 0.33 m, so we can substitute these values into the formula to find I.
4. Change in rotational energy (ΔE):
The change in rotational energy is equal to the work done on the disk. In this case, the disk starts from rest, so the initial rotational energy is zero. The final rotational energy can be calculated using the formula:
E = (1/2) * I * ω²
where I is the moment of inertia and ω is the final angular velocity.
To find the change in rotational energy, we subtract the initial energy from the final energy.
5. Tangential component of acceleration at half the final angular speed:
When the disk has reached half its final angular speed, we can calculate the tangential component of acceleration using the formula:
at = R * α
where R is the radius and α is the angular acceleration. Plug in the values to get the answer.
6. Magnitude of radial component of acceleration at half the final angular speed:
When the disk has reached half its final angular speed, the radial component of acceleration is zero since the acceleration is only tangential. So, the magnitude of the radial component of acceleration is zero.
7. Final speed at a point half-way between the center and the rim:
The final speed can be calculated using the formula:
v = R * ω
where R is the radius and ω is the angular velocity. In this case, R = 0.33 m and ω = 34 rad/s, so substitute these values into the formula to find v.
8. Total distance traveled by a point on the rim during 18 seconds:
To find the total distance, we can multiply the circumference of the disk (2πR) by the number of revolutions. The number of revolutions can be calculated by dividing the angular displacement by 2π.
So, multiply 2πR by (θ / 2π), where R is the radius and θ is the angular displacement obtained earlier.
By following these steps, we can find the answers to all the questions.