You are installing a new spark plug in your car, and the manual specifies that it be tightened to a torque that has a magnitude of 45.8 N*m. Using the data in the figure below (L = 0.287 m and θ = 49.6°), determine the magnitude F of the force that you must exert on the wrench.
To determine the magnitude F of the force that you must exert on the wrench, we can use the concept of torque. Torque is given by the equation:
Torque = Force * Lever Arm
In this case, the torque value is given as 45.8 N*m, and the lever arm is represented by L = 0.287 m. The angle between the force exerted and the lever arm is given as θ = 49.6°.
To find the magnitude of the force F, we rearrange the torque equation as follows:
Force = Torque / Lever Arm
First, let's convert the angle θ from degrees to radians by using the conversion factor: 1 radian = (π/180) degrees.
θ (in radians) = 49.6° * (π/180) ≈ 0.865 radians
Now, we can use the torque equation to calculate the magnitude of the force F:
F = 45.8 N*m / 0.287 m * sin(0.865 radians)
Substituting the given values into the equation:
F ≈ 45.8 N*m / 0.287 m * sin(0.865) ≈ 285.1 N
Therefore, the magnitude of the force F that you must exert on the wrench is approximately 285.1 N.