A wheel starts from rest and rotates about its axis with constant angular acceleration. After 8.7 s have elapsed, it has rotated through an angle of 20 radians. I found that the angular velocity at 8.7 s is 4.60 rad/s.
What is the angular acceleration of the wheel?
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?
To find the angular acceleration of the wheel, we can use the formula:
Δθ = ω_0 * t + (1/2) * α * t^2
Where:
Δθ - angle rotated (20 radians)
ω_0 - initial angular velocity (0 rad/s, since the wheel starts from rest)
t - time elapsed (8.7 s)
α - angular acceleration (what we need to find)
Let's rearrange the formula to solve for α:
α = (2 * Δθ - 2 * ω_0 * t) / t^2
Substituting the given values:
α = (2 * 20 radians - 2 * 0 rad/s * 8.7 s) / (8.7 s)^2
α = (40 radians) / (75.69 s^2)
α ≈ 0.529 rad/s^2
Therefore, the angular acceleration of the wheel is approximately 0.529 rad/s^2.
To find the centripetal acceleration of a point on the wheel at a distance r = 0.25 m from the axis at t = 8.7 s, we can use the formula:
a_c = r * α
Where:
a_c - centripetal acceleration
r - radius of rotation (0.25 m)
α - angular acceleration (0.529 rad/s^2, as found previously)
Substituting the values:
a_c = (0.25 m) * (0.529 rad/s^2)
a_c ≈ 0.132 m/s^2
Therefore, the centripetal acceleration of a point on the wheel at t = 8.7 s, located at a distance of 0.25 m from the axis, is approximately 0.132 m/s^2.