The table that follows lists four pairs of initial and final angles of a wheel on a moving car. The elapsed time for each pair of angles is 1.5 s. For each of the four pairs, determine the average angular velocity (magnitude and direction as given by the algebraic sign of your answer).
Initial angle θ0
Final angle θ
(a)
0.50 rad
0.76 rad
(b)
0.82 rad
0.52 rad
(c)
5.0 rad
3.6 rad
(d)
2.6 rad
4.0 rad
(a) Average angular velocity = 0.26 rad/s (clockwise)
(b) Average angular velocity = -0.30 rad/s (counterclockwise)
(c) Average angular velocity = -0.80 rad/s (counterclockwise)
(d) Average angular velocity = 0.70 rad/s (clockwise)
Oh, wheelie interesting question! Let's calculate the average angular velocity for each pair:
(a) For the first pair, the change in angle is 0.76 rad - 0.50 rad = 0.26 rad. Dividing this by the time of 1.5 s, we get an average angular velocity of approximately 0.17 rad/s. The positive sign indicates a clockwise direction.
(b) For the second pair, the change in angle is 0.52 rad - 0.82 rad = -0.30 rad (negative because it goes counterclockwise). Dividing this by 1.5 s, we get an average angular velocity of -0.20 rad/s.
(c) For the third pair, the change in angle is 3.6 rad - 5.0 rad = -1.4 rad (negative because it goes counterclockwise). Dividing this by 1.5 s, we get an average angular velocity of -0.93 rad/s.
(d) For the fourth pair, the change in angle is 4.0 rad - 2.6 rad = 1.4 rad. Dividing this by 1.5 s, we get an average angular velocity of approximately 0.93 rad/s. The positive sign indicates a clockwise direction.
Remember, angular velocity tells us about the rotation of an object. So, whether it's going clockwise or counterclockwise, at least we know it's spinning around and having a wheelie fun time!
To find the average angular velocity for each pair, we need to use the formula:
Average angular velocity = (final angle - initial angle) / elapsed time
Let's calculate the average angular velocity for each pair:
(a)
Initial angle θ0 = 0.50 rad
Final angle θ = 0.76 rad
Elapsed time = 1.5 s
Average angular velocity = (0.76 rad - 0.50 rad) / 1.5 s
Average angular velocity = 0.26 rad / 1.5 s
Average angular velocity = 0.173 rad/s
The average angular velocity for pair (a) is 0.173 rad/s.
(b)
Initial angle θ0 = 0.82 rad
Final angle θ = 0.52 rad
Elapsed time = 1.5 s
Average angular velocity = (0.52 rad - 0.82 rad) / 1.5 s
Average angular velocity = -0.30 rad / 1.5 s
Average angular velocity = -0.200 rad/s
The average angular velocity for pair (b) is -0.200 rad/s.
(c)
Initial angle θ0 = 5.0 rad
Final angle θ = 3.6 rad
Elapsed time = 1.5 s
Average angular velocity = (3.6 rad - 5.0 rad) / 1.5 s
Average angular velocity = -1.4 rad / 1.5 s
Average angular velocity = -0.933 rad/s
The average angular velocity for pair (c) is -0.933 rad/s.
(d)
Initial angle θ0 = 2.6 rad
Final angle θ = 4.0 rad
Elapsed time = 1.5 s
Average angular velocity = (4.0 rad - 2.6 rad) / 1.5 s
Average angular velocity = 1.4 rad / 1.5 s
Average angular velocity = 0.933 rad/s
The average angular velocity for pair (d) is 0.933 rad/s.
Therefore, the average angular velocities for the four pairs are as follows:
(a) 0.173 rad/s
(b) -0.200 rad/s
(c) -0.933 rad/s
(d) 0.933 rad/s
To find the average angular velocity, we can use the formula:
Average Angular Velocity (ω) = (θ - θ0) / Δt
where θ is the final angle, θ0 is the initial angle, and Δt is the elapsed time.
(a) For the first pair of angles:
Initial angle (θ0) = 0.50 rad
Final angle (θ) = 0.76 rad
Elapsed time (Δt) = 1.5 s
Average Angular Velocity (ω) = (0.76 rad - 0.50 rad) / 1.5 s
= 0.26 rad / 1.5 s
= 0.173 rad/s
The average angular velocity in this case is 0.173 rad/s. Since the final angle is greater than the initial angle, the direction of the angular velocity is counterclockwise.
You can calculate the average angular velocity for the remaining pairs of angles using the same formula:
(b) For the second pair of angles:
Initial angle (θ0) = 0.82 rad
Final angle (θ) = 0.52 rad
Elapsed time (Δt) = 1.5 s
Average Angular Velocity (ω) = (0.52 rad - 0.82 rad) / 1.5 s
= -0.3 rad / 1.5 s
= -0.2 rad/s
The average angular velocity in this case is -0.2 rad/s. Since the final angle is less than the initial angle, the direction of the angular velocity is clockwise.
(c) For the third pair of angles:
Initial angle (θ0) = 5.0 rad
Final angle (θ) = 3.6 rad
Elapsed time (Δt) = 1.5 s
Average Angular Velocity (ω) = (3.6 rad - 5.0 rad) / 1.5 s
= -1.4 rad / 1.5 s
= -0.933 rad/s
The average angular velocity in this case is -0.933 rad/s. Since the final angle is less than the initial angle, the direction of the angular velocity is clockwise.
(d) For the fourth pair of angles:
Initial angle (θ0) = 2.6 rad
Final angle (θ) = 4.0 rad
Elapsed time (Δt) = 1.5 s
Average Angular Velocity (ω) = (4.0 rad - 2.6 rad) / 1.5 s
= 1.4 rad / 1.5 s
= 0.933 rad/s
The average angular velocity in this case is 0.933 rad/s. Since the final angle is greater than the initial angle, the direction of the angular velocity is counterclockwise.