A penny farthing is a bicylce that was popular between 1870 and 1890. As the drawing shows, this type of bicycle has a large front wheel (R=1.20m) and a smaller rear wheel (r=.34m). A bicyclist, riding at a linear velocity of 12.5 m/s, applies the brake and produces an angular deceleration of 2.0 rad/s^2 in the front wheel. a)What is the angular displacement of the front and rear wheels? b)What is the average angular deceleration of the rear wheel during this motion?

To find the angular displacement of the front and rear wheels, we can use the equations of rotational motion.

a) Angular Displacement of the Front Wheel:
The angular deceleration is given as 2.0 rad/s^2.
The initial angular velocity of the wheel is 12.5 m/s (linear velocity) divided by the radius of the wheel (1.20 m). Therefore, the initial angular velocity is approximately 10.42 rad/s.

Using the equation: ωf^2 = ωi^2 + 2αθ,
where ωf = final angular velocity,
ωi = initial angular velocity,
α = angular acceleration, and
θ = angular displacement,

we can rearrange the equation to find the angular displacement:

θ = (ωf^2 - ωi^2) / (2α)

Since the final angular velocity is 0 rad/s (the wheel comes to a stop), the equation simplifies to:

θ = -ωi^2 / (2α)

Plugging in the values:
θ = -(10.42 rad/s)^2 / (2 * 2.0 rad/s^2)
= -107.8 rad

Therefore, the angular displacement of the front wheel is approximately -107.8 radians.

The negative sign indicates that the wheel rotates in the opposite direction (counterclockwise) as compared to its initial direction.

Angular Displacement of the Rear Wheel:
The same angular displacement applies to the rear wheel, as both wheels are rigidly connected and rotate together. Therefore, the angular displacement of the rear wheel is also approximately -107.8 radians.

b) Average Angular Deceleration of the Rear Wheel:
To find the average angular deceleration of the rear wheel, we can use the equation:

α_avg = (ωf - ωi) / Δt

Here, the initial angular velocity of the rear wheel is 10.42 rad/s, and the final angular velocity is 0 rad/s.
The change in time is not given in the question, so we cannot determine the exact average angular deceleration of the rear wheel without that information.

To solve this problem, we need to use the relationship between linear and angular quantities. The linear velocity v is related to the angular velocity ω by the equation v = ωr, where r is the radius of the wheel.

a) To find the angular displacement of the front wheel, we can use the equation ω = αt, where α is the angular acceleration and t is the time. Since the angular deceleration α is given as 2.0 rad/s^2, we can rearrange the equation to solve for time: t = ω/α.

Using the given linear velocity v = 12.5 m/s and the radius of the front wheel R = 1.20 m, we can find the angular velocity of the front wheel as follows:
ω = v/R.

Substituting this expression for ω in the equation for time, we have:
t = (v/R) / α.

Now we can calculate the angular displacement θ using the formula θ = ωt:
θ = (v/R) * (v/α).

Similarly, for the rear wheel, we have:
ω = v/r,
t = ω/α, and
θ = ωt.

Using the given radius of the rear wheel r = 0.34 m, we can perform the same calculations to find the angular displacement of the rear wheel.

b) The average angular deceleration of the rear wheel can be found by calculating the change in angular velocity Δω and dividing it by the time Δt.

Δω = final angular velocity - initial angular velocity = 0 - ω,
Δt = t (since the time taken to stop is the same for both wheels).

The average angular deceleration α_avg of the rear wheel is given by:
α_avg = Δω/Δt.

Now you can plug in the values and perform the calculations to find the answers.

a. Cf = pi*D = 3.14 * (2*1.2) = 7.54m =

Circumference of front wheel.

Vf=12.5m/s * 6.28rad / 7.54m=10.4rad/s.
= Angular velocity of front wheel.

Vf^2 = Vo^2 + 2ad = 0,
d = -(Vo)^2 / 2a = -(10.4)^2 / -4 = 27.04m = displacement of front wheel.

V = Vo + at = 0,
t = -Vo / a = -10.4 / -2 = 5.2s = time
required to stop.

a=(Vf-Vo)/t = -36.7 / 5.2 = -7.06m/s^2.

d = -(Vo)^2 / 2a = -(36.7)^2/-14.12 =
95.4m = displqcemnt of rear wheel.


Cr = 3.14 * (2*0.34) = 2.14m = circumference of rear wheel.

Vr=12.5m/s * 6.28rad/2.14m = 36.7rad/s
= Angular velocity of rear wheel.