A CD-Rom starts from rest and undergoes uniform angular acceleration until it reaches 26,000 rpm in 0.25 s. Through what angle does it rotate during this time?
f= 26,000 rpm=26000/3600=7.2 rev/s
2πf = εt
ε = 2πf/t =2π•7.2/0.25 =176 rad/s²
φ = εt²/2 = 176•(0.25)²/2 =5.5 rad
Oh, we've got ourselves a spinning CD-Rom, huh? Well, get ready for a wild ride with angles and rotation!
To find the angle of rotation, we need to use some angular kinematics equations. The formula we're looking for is:
θ = ωi * t + 0.5 * α * t^2
Where:
θ is the angle of rotation
ωi is the initial angular velocity (in radians per second)
t is the time (in seconds)
α is the angular acceleration (in radians per second squared)
Now, let's plug in the values we know. The initial angular velocity is zero (since it starts from rest), the time is 0.25 seconds, and the final angular velocity (ωf) is 26,000 rpm.
But, hold on a second! We need to convert that final angular velocity from rpm to radians per second. A quick conversion factor tells us there are 2π radians in one revolution, and 60 seconds in one minute. So:
26,000 rpm * (2π radians / 1 revolution) * (1 minute / 60 seconds) ≈ 2718.08 radians per second
Alright, now we can put all the pieces together and calculate the angle of rotation:
θ = (0 rad/s) * (0.25 s) + 0.5 * α * (0.25 s)^2
Since it starts from rest, ωi = 0, and we can simplify the equation to:
θ = 0.5 * α * (0.25 s)^2
Now, we just need to calculate the angular acceleration (α). We know that the final angular velocity is 2718.08 radians per second, and the initial angular velocity is 0. So:
α = (ωf - ωi) / t
= (2718.08 rad/s - 0 rad/s) / 0.25 s
= 10872.32 rad/s^2
Plugging this value back into the equation, we get:
θ = 0.5 * 10872.32 rad/s^2 * (0.25 s)^2
≈ 0.679 radians
So, during this time, the CD-ROM rotates about 0.679 radians. That's quite a spin for a little CD, isn't it?
To determine the angle of rotation, we need to find the final angular velocity and the initial angular velocity.
Given:
Initial angular velocity, ω₁ = 0 (since it starts from rest)
Final angular velocity, ω₂ = 26,000 rpm
Converting ω₂ from rpm to rad/s:
ω₂ = 26,000 rpm * (2π rad/1 min) * (1 min/60 s)
ω₂ = 26,000 * (2π/60) rad/s
ω₂ = 2,730.75 rad/s
Using the equation for angular acceleration:
ω₂ = ω₁ + α * t
2,730.75 rad/s = 0 + α * 0.25 s
Solving for α, the angular acceleration:
α = (2,730.75 rad/s) / (0.25 s)
α = 10,923 rad/s²
To find the angle of rotation, we use the equation relating angular displacement, initial angular velocity, final angular velocity, and angular acceleration:
θ = ω₁ * t + (1/2) * α * t²
Substituting the values:
θ = 0 * 0.25 s + (1/2) * (10,923 rad/s²) * (0.25 s)²
θ = 0 + (1/2) * 10,923 rad/s² * 0.0625 s²
θ = 0 + 0.5 * 682 rad
θ = 341 rad
Therefore, the CD-Rom rotates approximately 341 radians during this time.
To find the angle of rotation of the CD-ROM during this time, we need to use the formula for angular displacement, given by:
θ = ω_i * t + (1/2) * α * t^2
where:
θ is the angle of rotation,
ω_i is the initial angular velocity (in radians per second),
t is the time (in seconds), and
α is the angular acceleration (in radians per second squared).
In this case, the CD-ROM starts from rest, so ω_i is 0. Also, we are given the final angular velocity and the time. We need to find α first using the formula:
ω_f = ω_i + α * t
where:
ω_f is the final angular velocity.
In this case, ω_f is given as 26,000 rpm. To convert this to radians per second, we use the conversion factor:
1 rpm = (2π/60) radians per second
Substituting the given values:
26000 rpm = 26000 * (2π/60) radians per second
Simplifying, we get:
26000 rpm = 868.57 radians per second
Now we can find α:
ω_f = ω_i + α * t
868.57 = 0 + α * 0.25
Simplifying, we get:
α = 868.57 / 0.25
α = 3474.28 radians per second squared
Now that we have the value of α, we can compute θ:
θ = ω_i * t + (1/2) * α * t^2
θ = 0 * 0.25 + (1/2) * 3474.28 * 0.25^2
Simplifying, we get:
θ = 0.0859 radians
Therefore, the CD-ROM rotates by approximately 0.0859 radians during this time.