starting from rest, a disk rotates about its central axis with constant angular acceleration. In 4 seconds, it has rotated 50 rad.
What was the angualr acceleration during this time?
What is the instantaneous angular velocity of the disk at the end of the 4 seconds?
also, assuming that the acceleration does not change, through what additional angle will the disk turn during the next 7 seconds?
To find the angular acceleration, we can use the following kinematic equation:
θ = ω₀t + (1/2)αt²
where:
θ is the angle rotated in radians (given as 50 rad)
ω₀ is the initial angular velocity (given as 0 rad/s because it starts from rest)
t is the time taken (given as 4 s)
α is the angular acceleration (what we need to find)
Plugging in the values, we have:
50 rad = 0 rad/s * 4 s + (1/2)α * (4 s)²
Simplifying the equation:
50 rad = 8α s²
Now, solve for α:
α = (50 rad) / (8 s²)
α ≈ 6.25 rad/s²
So, the angular acceleration during this time is approximately 6.25 rad/s².
To find the instantaneous angular velocity at the end of the 4 seconds, we can use another kinematic equation:
ω = ω₀ + αt
where:
ω is the instantaneous angular velocity (what we need to find)
ω₀ is the initial angular velocity (0 rad/s)
α is the angular acceleration (6.25 rad/s² from the previous calculation)
t is the time taken (4 s)
Plugging in the values, we have:
ω = 0 rad/s + 6.25 rad/s² * 4 s
ω = 25 rad/s
So, the instantaneous angular velocity of the disk at the end of the 4 seconds is 25 rad/s.