Question

Determine the torque applied to the shaft of a car that transmits 450 hp and rotates at a rate of 3000 rpm.

Short answer

Answer: The torque applied to the shaft of the car is approximately 1070.9 N·m.

Step-by-step solution

Convert the power from horsepower to watts

To find the torque, we first need to convert the power from horsepower (hp) to watts (W). We know that 1 horsepower is equal to 746 watts. Therefore, we multiply the given power by 746 to convert it to watts: Power (W) = 450 hp * 746 W/hp = 336300 W

Convert the rotational speed from RPM to radians per second

Next, we need to convert the given rotational speed from rotations per minute (RPM) to radians per second (rad/s). The conversion factor is \(2\pi\) radians per rotation and 60 seconds per minute: Rotational speed (rad/s) = 3000 RPM * \(\frac{2\pi}{60}\) rad/s = 314.16 rad/s

Calculate the torque using the power-torque relationship

The relationship between power (W), torque (N·m), and rotational speed (rad/s) is given by the following formula: Power = Torque × Rotational speed To find the torque, we need to rearrange the formula: Torque = \(\frac{Power}{Rotational \thinspace speed}\) Now, we can substitute the values for Power and Rotational speed we calculated in previous steps: Torque = \(\frac{336300 W}{314.16 \thinspace \text{rad/s}}\) = 1070.9 N·m

Final answer

The torque applied to the shaft of the car is approximately 1070.9 N·m.