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 è = 0. Calculate the linear position 8 s after the wheel starts rotating. (Let the initial position of the point be 0.5 .)
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.)
We can't see your figure. There is no point posting such questions here.
To solve this problem, we need to analyze the given graph and use the formulas related to angular motion.
(a) The graph shows the angular acceleration versus time. To find the angular separation between the initial position and the point's position after 8 seconds, we need to integrate the angular acceleration graph to obtain the angular velocity.
1. Calculate the area under the graph from 0 to 8 seconds. This will give us the change in angular velocity.
2. To find the angular separation, use the formula:
Angular Separation = Initial angular position + (Initial angular velocity * time) + (0.5 * angular acceleration * time^2)
Since the flywheel is initially at rest, the initial angular velocity is 0.
(b) To find the linear position, velocity, and tangential acceleration, we need to use the relationship between linear and angular quantities:
1. Linear position = Angular position * radius
The radius of the flywheel is given as 1 meter.
2. Linear velocity = Angular velocity * radius
3. Tangential acceleration = Angular acceleration * radius
Now let's calculate the solutions step by step.
(a) To find the angular separation:
1. Calculate the area under the graph:
Since we have a graph, we can find the area by counting the total number of squares above and below the x-axis.
Let's assume that each square represents an interval of 1 second and calculate the total area.
Counting the squares, there are 4 squares above the x-axis and 5 squares below the x-axis.
Area = (4 * 1) + (5 * -1) = 4 - 5 = -1 rad/s
2. Calculate the angular separation:
Initial angular position = 0 (given)
Initial angular velocity = 0 (since the flywheel starts from rest)
Time = 8 s
Angular Separation = 0 + (0 * 8) + (0.5 * -1 * 8^2) = 0 - 32 = - 32 rad
Therefore, the angular separation between the initial position and the point's position 8 s after the wheel starts rotating is -32 rad.
(b) To find the linear position, velocity, and tangential acceleration:
1. Linear position = Angular position * radius
Given the angular separation as -32 rad and the radius as 1 m:
Linear position = -32 * 1 = -32 m
2. Linear velocity = Angular velocity * radius
Given that the angular velocity is 0 (since initial angular velocity is 0):
Linear velocity = 0 * 1 = 0 m/s
3. Tangential acceleration = Angular acceleration * radius
Given that the angular acceleration is constant at -1 rad/s^2 and the radius is 1 m:
Tangential acceleration = -1 * 1 = -1 m/s^2
Therefore, the linear position is -32 m, the linear velocity is 0 m/s, and the tangential acceleration is -1 m/s^2.