Question

An accelerometer is placed on the spoke of a wheel at a fixed radius r = 6.1 in. from the axle. As the bike travels at a constant speed, the accelerometer reads a constant value of a = 42.7 ft/s^2. If the bike wheel diameter is D = 21 in., what speed is the bike traveling at?

Answer 1

The speed of the bike traveling at a constant speed with an accelerometer reading of 42.7 ft/s^2 placed on a spoke 6.1 inches from the axle is approximately 3.18 miles per hour.

Step-by-step explanation:

To find the speed at which the bike is traveling, we can use the accelerometer reading, which tells us the centripetal acceleration experienced by the accelerometer placed on the wheel. Since centripetal acceleration (a) is given by the equation a = v2 / r, where v is the linear velocity and r is the radius through which the accelerometer is moving, we can rearrange this equation to solve for velocity as v = √(ar).

Firstly, we need to convert the given radius from inches to feet, because the acceleration is in feet per second squared. The radius (r) is 6.1 inches, which is r = 6.1 in × (1 ft / 12 in) = 0.5083 ft.

Now, we insert the values of acceleration (a = 42.7 ft/s2) and radius (r = 0.5083 ft) into the rearranged equation to find the linear velocity:

v = √(42.7 ft/s2 × 0.5083 ft) = √(21.72 ft2/s2) ≈ 4.66 ft/s

Finally, to find the speed of the bike, we need to convert this velocity back to a linear measurement that makes sense in the context of biking, such as miles per hour:

v = 4.66 ft/s × (3600 s/hr) × (1 mi / 5280 ft) = ≈ 3.18 mph.

Therefore, the bicycle is traveling at approximately 3.18 miles per hour.