To find the angular speed in rpm, we need to use the formula for tangential speed, which is the speed at which an object moves along a circular path. The formula for tangential speed is:
v = r * omega
where v is the tangential speed, r is the radius of the circle, and omega is the angular speed.
In this case, we are given that v = 25 mph and r = 15 feet. We can solve for omega by dividing both sides of the equation by r:
omega = v / r
Substituting the given values, we get:
omega = 25 mph / 15 feet
To convert the units of speed from mph to feet per second, we need to use the conversion factor that 1 mph is equal to 5280/3600 feet per second. We can use this conversion factor to rewrite the expression for omega in terms of feet per second:
omega = (25 mph * (5280/3600)) / 15 feet
This simplifies to:
omega = 0.7222 feet/second
Next, we need to convert the angular speed from feet per second to rpm. To do this, we need to use the formula for converting from linear speed to angular speed, which is:
omega = (v * 60) / (2 * pi * r)
where v is the linear speed, r is the radius of the circle, and omega is the angular speed in rpm.
In this case, we are given that v = 0.7222 feet/second and r = 15 feet. We can solve for omega by plugging these values into the formula and simplifying:
omega = (0.7222 feet/second * 60) / (2 * pi * 15 feet)
= 0.7222 feet/second * 60 / (2 * pi * 15 feet)
= 0.7222 * 60 / (2 * pi * 15)
= 4.3332 / (2 * pi * 15)
= 0.2821 / (pi * 15)
Therefore, the angular speed in rpm is 0.2821 / (pi * 15), or approximately 0.0046 rp