Suppose a car runs over a nail while driving at a speed of 54 miles per hour, and the nail is lodged in the tire tread 13 inches from the center of the wheel. what is the angular velocity of the nail in radians per hour?

The equation relating angular velocity and linear velocity is:

w = v / r

where,

w = angular velocity, v = linear velocity, and r = radius

Since r is in inches, convert v to inches:

v = (54 miles per hour) * (63360 inches / mile)

v = 3,421,440 inches / hour

Solving for w:

w = (3,421,440 inches / hour) / 13 inches

w = 263,187.70 rad / hour