A rock stuck in the tread of a 56.0--diameter bicycle wheel has a tangential speed of 3.10 . When the brakes are applied, the rock's tangential deceleration is 1.30m/s2

What is the magnitudes of the rock's angular velocity at = 1.70 ?

To find the magnitude of the rock's angular velocity at a given time, we can use the formula:

angular velocity (ω) = tangential speed (v) / radius (r)

First, let's convert the diameter of the bicycle wheel to radius by dividing it by 2:
radius (r) = diameter / 2 = 56.0 / 2 = 28.0 cm

Next, we need to convert the tangential speed from meters per second to centimeters per second:
tangential speed (v) = 3.10 m/s × 100 cm/m = 310 cm/s

Now, we have all the necessary values to calculate the angular velocity:
ω = v / r = 310 cm/s / 28.0 cm = 11.0714 rad/s

However, to find the angular velocity at a specific time (θ), we need to take into account the angular deceleration (α). We can use the following equation:

ω = ω0 + αθ

Given that the tangential deceleration (a) of the rock is 1.30 m/s^2, we can find the angular deceleration using the formula:

α = a / r

α = 1.30 m/s^2 / 28.0 cm = 0.0464 rad/s^2

Now, we can use the given time (θ = 1.70 s) and the angular deceleration to find the change in angular velocity:

ω = ω0 + αθ
ω = 11.0714 rad/s - 0.0464 rad/s^2 × 1.70 s = 11.0714 rad/s - 0.0790 rad/s = 10.9924 rad/s

Therefore, the magnitude of the rock's angular velocity at θ = 1.70 s is approximately 10.9924 rad/s.