A 44 m diameter wheel accelerates uniformly about its center from 110 rpm to 320 rpm in 3.9 s.
Determine the radial component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating.
To find the radial component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating, we can use the formula for linear acceleration:
a = Δv / Δt
where a is the linear acceleration, Δv is the change in velocity, and Δt is the change in time.
First, we need to calculate the change in velocity (Δv) of the point on the edge of the wheel. We are given the initial and final angular velocities, so we need to convert these into linear velocities using the formula:
v = ω * r
where ω is the angular velocity and r is the radius of the wheel.
Given:
Initial angular velocity (ω1) = 110 rpm
Final angular velocity (ω2) = 320 rpm
Radius of the wheel (r) = (44 m / 2) = 22 m
Convert ω1 and ω2 into linear velocities:
v1 = ω1 * r
v1 = (110 rpm) * (22 m) * (2π rad/60 s)
v1 = 154 m/s
v2 = ω2 * r
v2 = (320 rpm) * (22 m) * (2π rad/60 s)
v2 = 453.33 m/s
Next, we can calculate the change in velocity (Δv):
Δv = v2 - v1
Δv = 453.33 m/s - 154 m/s
Δv = 299.33 m/s
Now, we need to find the change in time (Δt) for 1.0 s:
Δt = 1.0 s
Finally, we can calculate the radial component of the linear acceleration (a) using the formula:
a = Δv / Δt:
a = (299.33 m/s) / (1.0 s)
a = 299.33 m/s^2
Therefore, the radial component of the linear acceleration of a point on the edge of the wheel 1.0 s after it has started accelerating is 299.33 m/s^2.