A \(3.0-\mathrm{kg}\) broom is leaning against a coffee table. \(A\) woman lifts the broom handle with her arm fully stretched so that her hand is a distance of \(0.45 \mathrm{~m}\) from her shoulder What torque is produced on her shoulder if her arm is at an angle of \(50^{\circ}\) below the horizontal? a) \(7.0 \mathrm{~N} \mathrm{~m}\) c) \(8.5 \mathrm{~N} \mathrm{~m}\) b) \(5.8 \mathrm{~N} \mathrm{~m}\) d) \(10.1 \mathrm{~N} \mathrm{~m}\)

Torque is a measure of how much a force acting on an object causes that object to rotate. The torque formula is a crucial concept in physics, particularly in mechanics, and it is given as:

\[\begin{equation}\tau = r \times F \times \sin(\theta)\end{equation}\]
Where

• \(\tau\) is the torque applied to the object,
• \(r\) is the distance from the pivot point to the point where the force is exerted,
• \(F\) is the magnitude of the force applied,
• \(\theta\) is the angle between the force vector and the lever arm.

In our example of the woman lifting a broom, the distance from her shoulder (pivot point) to her hand, the force, and the angle of her arm below the horizontal are all factors that determine the torque. It's important to remember that the sine of the angle is used to find the component of the force that causes rotation around the pivot point. Because the angle is critical in our calculation, it must be accurately converted into radians for the formula to work correctly.

The term 'weight' refers to the force exerted on a mass by the gravitational field of a massive body such as Earth. It's a common misconception that weight and mass are the same, but weight is actually a force and is measured in newtons, while mass is a property of an object and is measured in kilograms. To calculate the weight of an object, we use the formula:

\[\begin{equation}F = m \times g\end{equation}\]
Where

• \(F\) is the force of weight (in newtons),
• \(m\) is the mass of the object (in kilograms),
• \(g\) is the acceleration due to gravity (approximately \(9.81\mathrm{~m/s^2}\) on Earth).

When we applied this to our exercise involving the broom, we used the given mass of the broom, and multiplied by Earth's gravitational pull to arrive at the broom's weight. This weight is then used in the torque formula as the force component acting at a distance from the pivot point.

When working with angular measurements, it's important to understand the difference between degrees and radians. These are two units for measuring angles, with degrees being the more commonly known unit. However, in the realm of physics and engineering, radians are often preferred. The conversion between degrees and radians is a vital step in many calculations, including torque. The conversion formula from degrees to radians is:

\[\begin{equation}\theta_{\text{rad}} = \frac{\theta_{\text{deg}} \times \pi}{180}\end{equation}\]
This means that to convert an angle in degrees to radians, one multiplies the degree measure by \(\pi\) and then divides by 180. In the case of our example, the woman's arm creates an angle of 50 degrees below the horizontal; this angle in degrees was converted to radians before being used in the torque formula. Remember that angular conversion is essential because the trigonometric functions in the torque formula require angle measures in radians for correct calculation.