A flywheel with a diameter of 1 m is initially at rest. Its angular acceleration versus time is graphed in the figure.
(a) What is the angular separation between the initial position of a fixed point on the rim of the flywheel and the point's position 8 s after the wheel starts rotating? (Be sure to give angular separation rather than total angular displacement.)
(b) The point starts its motion at θ = 0. Calculate the linear position 8 s after the wheel starts rotating. (Let the initial position of the point be 0.5 x) m
Calculate the linear velocity vector 8 s after the wheel starts rotating. (Enter the magnitude.)
Calculate the tangential acceleration vector after the wheel starts rotating. (Enter the magnitude.)
I have no idea what angular acceleration is.
To solve this problem, we need to use the given angular acceleration versus time graph and the properties of circular motion.
(a) The angular separation between the initial position and the position after 8 seconds can be found by integrating the angular acceleration versus time graph. Since the angular separation is the area under the graph, we can use the trapezoidal rule to approximate the integral.
First, divide the time interval into smaller intervals. Let's say we divide it into 4 intervals: 0-2 s, 2-5 s, 5-7 s, and 7-8 s.
Next, calculate the area of each trapezoid. For the first interval (0-2 s), the height is the angular acceleration at t=0 and the top base is the angular acceleration at t=2. Multiply the height by the average of the top and bottom bases to find the area. Repeat this process for the other intervals.
Finally, sum up the areas of all the trapezoids to find the approximate angular separation.
(b) To find the linear position 8 s after the wheel starts rotating, we can use the definition of angular displacement. Angular displacement (θ) is related to linear displacement (x) by the formula θ = x/r, where r is the radius of the flywheel.
Given that the initial position of the point is 0.5x and the angular separation at 8 s is obtained from part (a), we can calculate the linear position by using the formula x = θr.
(c) Linear velocity can be calculated using the formula v = ωr, where ω is the angular velocity. To find ω, we can integrate the angular acceleration graph from part (a) with respect to time to find the angular velocity versus time graph. Then, we can find the angular velocity at t=8 s and multiply it by the radius to get the linear velocity.
(d) The tangential acceleration can be found using the formula a_t = ω^2r. We can find ω from part (c) and multiply it by the radius to get the tangential acceleration.
By following these steps, we can find the answers to each part of the question.