Questions: The tune-up specifications of a car call for the spark plugs to be tightened to a torque of 41 N·m. You plan to tighten the plugs by pulling on the end of a 24-cm-long wrench. Because of the cramped space under the hood, you'll need to pull at an angle of 110° with respect to the wrench shaft. With what force must you pull? Express your answer in newtons.

Solution Steps

The problem involves calculating the force required to achieve a specified torque when tightening spark plugs using a wrench. The torque required is \(41 \, \text{N} \cdot \text{m}\), and the wrench is \(24 \, \text{cm}\) long. The force is applied at an angle of \(110^\circ\) with respect to the wrench shaft.

Convert the length of the wrench from centimeters to meters for consistency in units: \[ 24 \, \text{cm} = 0.24 \, \text{m} \]

The torque (\(\tau\)) is given by the formula: \[ \tau = r \cdot F \cdot \sin(\theta) \] where:

• \(\tau = 41 \, \text{N} \cdot \text{m}\)
• \(r = 0.24 \, \text{m}\)
• \(\theta = 110^\circ\)

Rearrange the formula to solve for the force \(F\): \[ F = \frac{\tau}{r \cdot \sin(\theta)} \]

Substitute the known values into the equation: \[ F = \frac{41}{0.24 \cdot \sin(110^\circ)} \]

Calculate \(\sin(110^\circ)\): \[ \sin(110^\circ) \approx 0.9397 \]

Substitute \(\sin(110^\circ)\) into the equation for \(F\): \[ F = \frac{41}{0.24 \cdot 0.9397} \approx \frac{41}{0.2255} \approx 181.8 \, \text{N} \]

Final Answer