The crankshaft in your car’s engine rotates at a rate of 2,400 revolutions per minute (RPMs). What is the angular velocity of the crankshaft in SI units of rad/s?
The Correct Answer and Explanation is:
To find the angular velocity of the crankshaft in SI units of rad/s, we need to convert from revolutions per minute (RPM) to radians per second (rad/s).
- Understanding the Conversion:
- One complete revolution corresponds to an angle of 2π2\pi2π radians.
- Since the crankshaft rotates at 2,400 revolutions per minute (RPM), this means it completes 2,400 full rotations in one minute.
- Converting RPM to Radians per Minute (rad/min):
- For each revolution, the crankshaft covers 2π2\pi2π radians.
- Therefore, in 2,400 RPM, the total angle covered in one minute is: 2,400×2π rad/min=4,800π rad/min.2,400 \times 2\pi \text{ rad/min} = 4,800\pi \text{ rad/min}.2,400×2π rad/min=4,800π rad/min.
- Converting Minutes to Seconds:
- There are 60 seconds in a minute, so we divide by 60 to convert from rad/min to rad/s: 4,800π rad/min60=80π rad/s.\frac{4,800\pi \text{ rad/min}}{60} = 80\pi \text{ rad/s}.604,800π rad/min=80π rad/s.
- Final Answer: Angular velocity=80π rad/s.\text{Angular velocity} = 80\pi \text{ rad/s}.Angular velocity=80π rad/s. Using the approximate value of π≈3.14159\pi \approx 3.14159π≈3.14159, we get: 80π≈251.33 rad/s.80\pi \approx 251.33 \text{ rad/s}.80π≈251.33 rad/s.
Explanation:
The crankshaft in your car’s engine rotates at 2,400 RPM, meaning it makes 2,400 full rotations every minute. Each full rotation corresponds to 2π2\pi2π radians. By multiplying the number of rotations per minute (2,400) by 2π2\pi2π, we calculate the total angular displacement per minute in radians. To convert this into SI units (radians per second), we divide by 60 (the number of seconds in a minute). This gives us the angular velocity in rad/s, which is approximately 251.33 rad/s.
Thus, the crankshaft’s angular velocity is about 251.33 rad/s.
