Final answer:
To find the number of revolutions per minute (rpm) of the front sprocket wheel that will produce a speed of 40 mi/hr, we first convert the speed to inches per second. Then, using the linear velocity formula, we calculate the angular velocity of the front sprocket wheel. Finally, we convert the angular velocity from rad/s to rpm to obtain the answer of approximately 448 rpm.
Step-by-step explanation:
To determine the number of revolutions per minute (rpm) of the front sprocket wheel that will produce a speed of 40 mi/hr, we need to convert the speed from miles per hour to inches per second. We know that 1 mile is equal to 5280 feet and 1 foot is equal to 12 inches, so 1 mile is equal to 63360 inches. Therefore, 40 mi/hr is equal to 40 * 63360 = 2534400 inches/hr. To convert to inches per second, we divide this value by 3600 (the number of seconds in an hour), giving us a speed of approximately 704 inches/sec.
Next, we need to find the linear velocity of the front sprocket wheel. This can be calculated using the formula v = rω, where v is the linear velocity, r is the radius of the wheel, and ω is the angular velocity in rad/s. Since we want to find ω (in rpm), we can rearrange the formula to ω = v / r. The radius of the front sprocket wheel is 5 inches, so the angular velocity is approximately 704 / 5 = 140.8 rad/s.
Finally, to convert the angular velocity from rad/s to rpm, we divide by 2π (the number of radians in a revolution) and multiply by 60 (the number of seconds in a minute). This gives us an angular velocity of approximately 140.8 * 60 / (2π) ≈ 448 rpm.