Question

A mechanic uses a wrench that is \(22 \mathrm{~cm}\) long to tighten a spark plug. If the mechanic exerts a force of \(58 \mathrm{~N}\) to the end of the wrench, what is the maximum torque she can apply to the spark plug?

Short answer

The maximum torque is 12.76 Nm.

Step-by-step solution

Understanding Torque

Torque is a measure of the force that can cause an object to rotate about an axis. The formula for torque (\(\tau\)) is given by:\[\tau = r \times F \times \sin(\theta)\]where \(r\) is the lever arm (distance from the pivot point to the point of force application), \(F\) is the force applied, and \(\theta\) is the angle between the force and lever arm. In this context, \(\theta = 90^\circ\) because the force is applied perpendicularly, so \(\sin(\theta) = 1\).

Lever Arm and Force Substitution

Given in this problem, the lever arm (\(r\)) is the length of the wrench, which is \(22 \mathrm{~cm}\) or \(0.22 \mathrm{~m}\) (converted from centimeters to meters for unit consistency), and the force (\(F\)) is \(58 \mathrm{~N}\). Since the angle is \(90^\circ\), the formula simplifies to:\[\tau = r \times F\]

Calculate the Torque

Substitute the values of the lever arm and force into the torque formula:\[\tau = 0.22 \mathrm{~m} \times 58 \mathrm{~N} = 12.76 \mathrm{~Nm}\]So, the maximum torque that can be applied to the spark plug is \(12.76 \mathrm{~Nm}\).

Key concepts

Lever Arm

In physics, the lever arm is a vital concept in understanding how forces cause an object to rotate. The lever arm, often denoted as \(r\), is the perpendicular distance from the axis of rotation (also called the pivot point) to the line of action of the force. The longer the lever arm, the greater the rotational effect or torque for the same amount of force.

In our exercise, the lever arm is the length of the wrench, which is crucial in determining torque. Always remember to convert measurements into consistent units, often meters, when calculating torque. Therefore, the wrench length of \(22 \text{ cm}\) is converted to \(0.22 \text{ m}\) to maintain unit consistency. This distance dictates how much of the applied force translates into torque, emphasizing the importance of measuring the lever arm accurately.

Force Application

Applying force at a strategic angle is crucial in torque calculations. The force, represented by \(F\), is the external push or pull exerted on an object.

In our exercise, the force applied by the mechanic is \(58 \text{ N}\). When applied at a 90-degree angle to the lever arm, the force maximizes its effectiveness in producing torque, since the sine of 90 degrees is 1. Therefore, in calculations where the angle is exactly 90 degrees, the torque formula simplifies to \(\tau = r \times F\). This simplification is practical for many everyday scenarios, such as using a wrench, where the force is deliberately applied perpendicularly to achieve maximum torque.

Physics Problem Solving

Solving physics problems, like calculating torque, involves understanding and applying fundamental principles accurately. Key to problem-solving is breaking down the question step-by-step to identify what is needed.

In the wrench problem, the process involves determining the lever arm, ascertaining the force applied, and recognizing the angle at which the force is applied. Each of these elements directly influences the torque calculation.

• First, convert measurements to consistent units.
• Second, apply the torque formula \(\tau = r \times F\), leveraging the angle's simplicity when \(\theta = 90^\circ\).
• Finally, substitute and calculate to find the torque.

With these steps, systematically work through physics problems ensuring a clear understanding of each component involved in the calculations.