91Ó°ÊÓ

Calculating force is essential when you're dealing with different physical applications, such as tightening a lug nut on a vehicle. In this exercise, we've encountered a situation where we need to adhere to a maximum torque, which is a measure of the force causing an object to rotate. Torque is represented by the formula:

• \( \tau = r \times F \times \sin(\theta) \)

Here, \( r \) is the length of the wrench (the lever arm), \( F \) is the force applied at the end of the wrench, and \( \theta \) is the angle at which force is applied relative to the lever arm.
To calculate the force exerted without exceeding the torque limit, we rearrange the torque formula to solve for force (\( F \)) as follows:

• \( F = \frac{\tau}{r \times \sin(\theta)} \)

By substituting the values like torque, the length of the wrench, and the appropriate angle, you can compute the specific force that can be safely applied. In this exercise, the calculated force turned out to be \( 226.29 \) N.

Angles are often presented in degrees, but most trigonometric functions in physics use radians. Therefore, converting degrees to radians is a common step in these calculations.
The conversion between degrees and radians uses the formula:

• \( \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \)

In this exercise, the angle given was \( 75.0^{\circ} \), which you converted to radians by multiplying by \( \frac{\pi}{180} \). This results in an angle of approximately \( 1.31 \) radians.
Remembering this conversion factor is quite handy since many scientific calculators will require angle inputs in radians when performing trigonometric calculations.

Understanding trigonometric functions is essential to solving problems involving angles. These functions, such as sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), help describe the relationship between the angles and sides of a triangle.
In torque problems, the sine function is particularly important. When a force is applied at an angle, it's only the component of the force that acts perpendicular to the lever arm which contributes to torque. This is why the torque formula uses \( \sin(\theta) \).

• \( \sin(\theta) \) represents the ratio of the opposite side over the hypotenuse in a right-angle triangle

By applying \( \sin(1.31) \) in the exercise, we find the effective force component that influences torque, allowing us to solve for the correct force magnitude without exceeding the limit. Always ensure angle measures are properly converted and applied in these calculations for accurate results.