A cyclist starts from rest and pedals such that the wheels of his bike have a constant angular acceleration. After 20.0 s, the wheels have made 219 rev. What is the angular acceleration of the wheels?

What is the angular velocity of the wheels after 20.0 s?

If the radius of the wheel is 34.0 cm, and the wheel rolls without slipping, how far has the cyclist traveled in 20.0 s?

Why did the cyclist join a circus? Because he wanted to wheel-y show off his angular acceleration skills!

To find the angular acceleration of the wheels, we can use the formula:
angular acceleration (α) = (final angular velocity - initial angular velocity) / time

Now, since the cyclist starts from rest, the initial angular velocity is zero. We are given that the wheels have made 219 rev (or revolutions) after 20.0 s. Since a revolution is 360 degrees, we can convert that to radians by multiplying by 2π.

So, the final angular velocity is (219 rev) * (2π rad/rev) / (20.0 s) = 21.9π rad/s.

Now, using the formula, α = (21.9π rad/s - 0 rad/s) / (20.0 s) = 1.095π rad/s².

The angular velocity after 20.0 s is given as 21.9π rad/s.

Now, to find the distance the cyclist has traveled, we can use the formula:
distance = (angular velocity) * (radius) * (time)

Plugging in the values, distance = (21.9π rad/s) * (0.34 m) * (20.0 s) = 147.60π m.

So, the cyclist has traveled approximately 147.60π m.

Keep on rollin' and laughing!

To find the angular acceleration of the wheels, we can use the formula:

θ = ω₀t + (1/2)αt²

where θ is the total angle covered, ω₀ is the initial angular velocity, t is the time, and α is the angular acceleration.

Given:
θ = 219 rev
t = 20.0 s

Converting the angle from rev to radians:
θ = 219 rev * (2π rad/1 rev) = 219 * 2π rad

Plugging in the values in the formula:
θ = ω₀t + (1/2)αt²
219 * 2π rad = ω₀ * 20.0 s + (1/2)α * (20.0 s)²

Simplifying and rearranging the equation:
11.0π rad = ω₀ * 20.0 s + 200.0α s²

We also know that the initial angular velocity ω₀ is 0 rad/s because the cyclist starts from rest:

11.0π rad = 0 rad/s * 20.0 s + 200.0α s²

Simplifying the equation, we get:
11.0π rad = 200.0α s²

Now, we can solve for α, the angular acceleration:
α = (11.0π rad) / (200.0 s²)

Using a calculator, we find:
α ≈ 0.173 rad/s²

Therefore, the angular acceleration of the wheels is approximately 0.173 rad/s².

To find the angular velocity of the wheels after 20.0 s, we can use the formula:

ω = ω₀ + αt

Given:
ω₀ = 0 rad/s (initial angular velocity)
t = 20.0 s

Plugging in the values in the formula:
ω = 0 rad/s + (0.173 rad/s²)(20.0 s)

Simplifying the equation, we get:
ω = 3.46 rad/s

Therefore, the angular velocity of the wheels after 20.0 s is 3.46 rad/s.

To find the distance traveled by the cyclist in 20.0 s, we can use the formula:

s = rθ

where s is the distance traveled, r is the radius of the wheel, and θ is the total angle covered.

Given:
r = 34.0 cm = 0.34 m
θ = 219 rev * (2π rad/1 rev) = 219 * 2π rad

Plugging in the values in the formula:
s = (0.34 m)(219 * 2π rad)

Simplifying the equation, we get:
s ≈ 478.6 m

Therefore, the cyclist has traveled approximately 478.6 meters in 20.0 seconds.

To find the angular acceleration of the wheels, we can use the formula:

angular acceleration = change in angular velocity / change in time

In this case, the cyclist starts from rest, so the initial angular velocity is zero. The final angular velocity can be found using the formula:

final angular velocity = initial angular velocity + (angular acceleration * time)

Since the cyclist starts from rest, the initial angular velocity is zero, and the time given is 20.0 s.

Using the given information, we can substitute the values into the formula to find the angular acceleration:

final angular velocity = 219 rev
time = 20.0 s

final angular velocity = initial angular velocity + (angular acceleration * time)
219 rev = 0 + (angular acceleration * 20.0 s)

By rearranging the equation, we can solve for the angular acceleration:

angular acceleration = (final angular velocity - initial angular velocity) / time
angular acceleration = (219 rev - 0 rev) / 20.0 s

Therefore, the angular acceleration of the wheels is:

angular acceleration = 10.95 rev/s^2

Next, let's find the angular velocity of the wheels after 20.0 s. We can use the same formula:

final angular velocity = initial angular velocity + (angular acceleration * time)

Again, the initial angular velocity is zero, and we know the angular acceleration is 10.95 rev/s^2 and the time is 20.0 s.

final angular velocity = 0 rev/s + (10.95 rev/s^2 * 20.0 s)

Simplifying the equation, we find:

final angular velocity = 219 rev/s

Therefore, the angular velocity of the wheels after 20.0 s is 219 rev/s.

Now, let's find the distance the cyclist has traveled in 20.0 s. Since the wheel rolls without slipping, the distance traveled is equal to the circumference of the wheel multiplied by the number of revolutions.

The formula for the distance traveled is:

distance = circumference * revolutions

Given that the radius of the wheel is 34.0 cm, the circumference of the wheel is:

circumference = 2 * π * radius
circumference = 2 * π * 34.0 cm

Now, we can calculate the distance traveled:

distance = circumference * revolutions
distance = (2 * π * 34.0 cm) * 219 rev

Simplifying the equation, we find:

distance = 46,082.3 cm

Therefore, the cyclist has traveled approximately 460.823 meters in 20.0 s.

Va=219rev/20s * 6.28rad/rev=68.8 rad/s

a=(V-Vo)/t = (68.8-0)/20=3.44 rad/s^2.

Circumference=pi*2r = 6.28 * 68=427 cm=
4.27 m.

d = 219revs * 4.27m/rev = 935.2 m.