Your roommate is working on his bicycle and has the bike upside down. He spins the 68.0cm -diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second.

And is there a question here?

To determine the speed at which the pebble is moving, we need to find the linear velocity of the wheel.

The linear velocity of a point on the perimeter of a rotating object can be calculated using the formula:

Linear velocity = circumference × angular velocity

To find the circumference, we can use the diameter of the wheel, which is given as 68.0 cm. The circumference can be calculated as:

Circumference = π × diameter

Plugging in the values, we have:

Circumference = π × 68.0 cm

Next, we need to determine the angular velocity of the wheel.

Angular velocity is the rate at which an object rotates, measured in radians per unit of time. In this case, we can calculate it using the number of revolutions per second.

Since the pebble goes by three times every second, it means the wheel completes three revolutions per second.

To convert revolutions per second to radians per second, we multiply by 2π, as there are 2π radians in one revolution.

Angular velocity = 3 rev/s × 2π rad/rev

Once we have both the circumference and angular velocity, we can calculate the linear velocity:

Linear velocity = Circumference × Angular velocity

Now let's plug in the values we obtained to find the result.