Questions: A mechanic changing the spark plugs in a car notes that the instruction manual calls for a torque with a magnitude of 43 N . m . If the mechanic grasps the wrench as shown in the figure below, determine the magnitude (in N ) of the force she must exert on the wrench.

A mechanic changing the spark plugs in a car notes that the instruction manual calls for a torque with a magnitude of 43 N . m . If the mechanic grasps the wrench as shown in the figure below, determine the magnitude (in N ) of the force she must exert on the wrench.

Transcript text: A mechanic changing the spark plugs in a car notes that the instruction manual calls for a torque with a magnitude of 43 N . m . If the mechanic grasps the wrench as shown in the figure below, determine the magnitude (in N ) of the force she must exert on the wrench. $\square$ N Additional Materials

Solution

Solution Steps

Step 1: Identify the Given Values

Torque ($ \tau $) = 43 N·m
Length of the wrench ($ r $) = 0.30 m
Angle ($ \theta $) = 60°

Step 2: Understand the Relationship Between Torque, Force, and Angle

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

$ \tau $ is the torque,
$ r $ is the length of the wrench,
$ F $ is the force applied,
$ \theta $ is the angle between the force and the wrench.

Step 3: Rearrange the Formula to Solve for Force

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

Step 4: Substitute the Given Values into the Formula

Substitute the given values into the formula: \[ F = \frac{43 \, \text{N·m}}{0.30 \, \text{m} \cdot \sin(60°)} \]

Step 5: Calculate the Sine of the Angle

Calculate $ \sin(60°) $: \[ \sin(60°) = \frac{\sqrt{3}}{2} \approx 0.866 \]

Step 6: Perform the Final Calculation

Substitute $ \sin(60°) $ into the formula and calculate $ F $: \[ F = \frac{43}{0.30 \cdot 0.866} \] \[ F \approx \frac{43}{0.2598} \] \[ F \approx 165.5 \, \text{N} \]