A wheel starts from rest and rotates about its axis with constant angular acceleration. After 8.7s have elapsed, it has rotated through an angle of 20 radians.
What is the angular acceleration of the wheel?
What is the angular velocity when the time t = 8.7 s? I got the answer of 4.60 rad/s
What is the centripetal acceleration of a point on the wheel a distance r = 0.25 m from the axis at t = 8.7 s?
20 radians = (1/2)alpha t^2
20 = (a/2) (8.7)^2
a = 40/8.7^2 = .528
v = a t = .528 * 8.7 = 4.60 rad/s yes
Ac = v^2/r = omega^2 r = 8.7^2 (.25)
Thank you, but the centripetal acceleration is wrong, 2.95 m/s^2 is wrong.
To find the angular acceleration of the wheel, we can use the formula:
θ = ω₀t + 0.5αt²
where:
θ is the angle rotated (20 radians in this case),
ω₀ is the initial angular velocity (which is zero since the wheel starts from rest),
t is the time elapsed (8.7 seconds in this case), and
α is the angular acceleration (what we're trying to find).
Plugging in the given values, we have:
20 radians = 0*(8.7s) + 0.5α*(8.7s)²
Solving for α:
20 radians = 0 + 0.5α*(75.69s²)
40 radians = α*(75.69s²)
α = 40 radians / 75.69s²
α ≈ 0.529 radians/s²
Therefore, the angular acceleration of the wheel is approximately 0.529 radians/s².
To find the angular velocity at t = 8.7s, we can use the formula:
ω = ω₀ + αt
Plugging in the given values:
ω = 0 + (0.529 rad/s²)*(8.7s)
ω ≈ 4.60 rad/s
Therefore, the angular velocity at t = 8.7s is approximately 4.60 rad/s.
To find the centripetal acceleration of a point on the wheel at r = 0.25m at t = 8.7s, we can use the formula:
a = rα
Plugging in the given values:
a = (0.25m)*(0.529 rad/s²)
a ≈ 0.1322 m/s²
Therefore, the centripetal acceleration of a point on the wheel at r = 0.25m at t = 8.7s is approximately 0.1322 m/s².
To find the angular acceleration of the wheel, we can use the formula:
θ = ω0 * t + (1/2) * α * t^2
where θ is the angle rotated, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
In this case, θ = 20 radians and t = 8.7 seconds. Since the wheel starts from rest, ω0 = 0. Plugging in the values, we get:
20 = 0 * 8.7 + (1/2) * α * (8.7^2)
Simplifying the equation, we get:
20 = (1/2) * α * 75.69
Now we can solve for α:
40 = α * 75.69
Dividing both sides by 75.69:
α = 40 / 75.69 ≈ 0.528 rad/s^2
So the angular acceleration of the wheel is approximately 0.528 rad/s^2.
To find the angular velocity (ω) at t = 8.7 s, we can use the formula:
ω = ω0 + α * t
Since ω0 = 0 (the wheel starts from rest) and α = 0.528 rad/s^2, we can calculate ω:
ω = 0 + 0.528 * 8.7
ω ≈ 4.60 rad/s
So the angular velocity of the wheel at t = 8.7 s is approximately 4.60 rad/s.
Now, to find the centripetal acceleration of a point on the wheel located at a distance r = 0.25 m from the axis at t = 8.7 s, we can use the formula:
a = r * ω^2
Plugging in the values, we get:
a = 0.25 * (4.60)^2
a ≈ 2.575 m/s^2
So the centripetal acceleration of the point on the wheel is approximately 2.575 m/s^2.