A cyclist accelerates from rest. After 7 s, the wheels have made 2 revolutions.
a. What is the angular acceleration of the wheels?
b. What is the angular velocity of the wheels after 7 s?
Note: we need to do part b first
b. The average angular velocity is 2*(2pi) rad / 7 s = 4*pi / 7 rad/s = 1.8 rad/s
That is the average. The question did not say, but we have assume the acceleration was constant. So the velocity, at t=7 s, is
2*1.8 rad/s = 3.6 rad/s
a. Now that we know angular velocity, w, at 7 s, angular acceleration, alpha, is
alpha = (change in w)/time = 3.6 rad/7 s = 0.51 rad/s^2
a. Well, if the wheels have made 2 revolutions in 7 seconds, it means they're definitely spinning faster than a politician trying to avoid answering a direct question! To find the angular acceleration, we can use the equation:
Angular acceleration (α) = (Change in angular velocity)/(Change in time)
Since the cyclist started from rest, the initial angular velocity (ω0) is 0. Then, let's calculate the final angular velocity (ω):
Number of revolutions = 2
→ 2π radians = ω * 7 seconds
Solving for ω, we get:
ω = 2π/7 rad/s
Now, we can determine the angular acceleration:
α = (ω - ω0)/(Change in time)
= (2π/7 - 0)/7
≈ 0.897 rad/s²
b. After 7 seconds, the angular velocity (ω) of the wheels will be approximately 2π/7 rad/s. You know, it's like the wheels are saying, "Hello, world! We're spinning our way into the record books!"
To find the angular acceleration of the wheels, we can use the formula:
Angular acceleration = Change in angular velocity / Time
Given that the cyclist starts from rest and completes 2 revolutions in 7 seconds, we can find the change in angular velocity as follows:
The number of revolutions can be converted to the number of complete circles made by the wheels:
2 revolutions = 2 * 2π radians = 4π radians
The time taken is 7 seconds.
So, the angular velocity after 7 seconds would be:
Angular velocity = Change in angle / Time = 4π radians / 7 seconds
We need to find the angular acceleration, so we rearrange the formula:
Angular acceleration = Change in angular velocity / Time = (4π radians / 7 seconds) / 7 seconds
Simplifying this expression, we get:
Angular acceleration = 4π / 7^2 radians per second squared
Therefore:
a. The angular acceleration of the wheels is 4π / 49 radians per second squared.
b. The angular velocity of the wheels after 7 seconds is 4π / 7 radians per second.
To determine the angular acceleration and angular velocity of the wheels, we can use the equations of rotational motion:
a. Angular acceleration (α) is defined as the rate of change of angular velocity (ω) over time (t). We can use the equation:
α = (ω - ω0) / t
where ω0 is the initial angular velocity, which is zero in this case since the cyclist starts from rest.
b. Angular velocity is defined as the rate of change of angular displacement (θ) over time (t). We can use the equation:
ω = θ / t
Given that the wheels have made 2 full revolutions, we can calculate the angular displacement as follows:
θ = 2πn
where n is the number of revolutions. In this case, n = 2.
Now, let's calculate the answers:
a. To find the angular acceleration:
First, we need to calculate the angular displacement:
θ = 2πn = 2π(2) = 4π radians
Next, substitute the values into the formula for angular acceleration:
α = (ω - ω0) / t
= (θ / t - ω0) / t
= (4π / 7 - 0) / 7
= 4π / 49 radians per second squared
Therefore, the angular acceleration of the wheels is 4π / 49 radians per second squared.
b. To find the angular velocity after 7 s:
Substitute the values into the formula for angular velocity:
ω = θ / t
= 4π / 7 radians per second
Therefore, the angular velocity of the wheels after 7 seconds is 4π / 7 radians per second.
a. V = 2rev/7s * 6.28rad/rev = 1.79 rad/s.
V = a * t = 1.79.
a*7 = 1.79,
a = 0.256 rad/s^2.
b. V = 1.79 rad/s.(see part a).