Question

To tighten a spark plug, it is recommended that a torque of \(15 \mathrm{~N} \cdot \mathrm{m}\) be applied. If a mechanic tightens the spark plug with a wrench that is \(25 \mathrm{~cm}\) long, what is the force necessary to create the desired torque?

Short answer

The necessary force is 60 N.

Step-by-step solution

Understand the Relationship

Torque (\(T\)) is the product of the force exerted (\(F\)) and the lever arm length (\(r\)) it acts on. This relationship can be expressed with the formula \(T = F \, \times \, r\). In this problem, \(T\) (torque) is \(15 \, \mathrm{N\cdot m}\), and \(r\) (the length of the wrench) is \(25 \, \mathrm{cm}\). To proceed, remember to convert the length to meters for consistency in units: \(25 \, \mathrm{cm} = 0.25 \, \mathrm{m}\).

Rearrange the Formula

To find the force, we need to solve for \(F\) in the equation \(T = F \, \times \, r\). Rearrange the equation to find \(F\) by dividing both sides by \(r\). This gives \(F = \frac{T}{r}\).

Substitute Values

Now substitute the given values into the equation: \(F = \frac{15 \, \mathrm{N\cdot m}}{0.25 \, \mathrm{m}}\).

Calculate the Force

Perform the division to find \(F\): \[F = \frac{15}{0.25} = 60 \, \mathrm{N}\]Thus, the force necessary is \(60 \, \mathrm{N}\).

Key concepts

Lever Arm

The concept of a lever arm is an essential part of understanding how torque works. It refers to the perpendicular distance from the axis of rotation to where the force is applied. The longer the lever arm, the more torque is produced for the same amount of force.

In our exercise, the lever arm is the wrench, which is 25 cm long. By converting this measurement to meters (0.25 m), we ensure all our units are consistent for calculations. This length of the wrench acts as the lever arm, allowing the mechanic to apply force effectively to produce torque.

When solving problems involving torque, always identify the length of the lever arm first, as it's crucial for determining how much force is needed.

Force Calculation

Force calculation plays a critical role in solving torque-related problems. The key to determining the required force lies in using the torque formula:

• \( T = F \times r \)

Where \( T \) is torque, \( F \) is the force, and \( r \) is the lever arm length.

In our scenario, we rearrange this formula to solve for force \( F \):

• \( F = \frac{T}{r} \)

This rearrangement allows us to calculate the force by dividing the torque by the lever arm length.

After substituting the given values—the torque of 15 Nm and the lever arm of 0.25 m—we find \( F = \frac{15}{0.25} = 60 \) N. This result means a force of 60 Newtons is necessary to achieve the desired torque.

Newton Meter

In physics, the unit of torque is the Newton meter (Nm), which combines the unit of force (Newton) with the unit of distance (meter). This unit expresses both the strength and displacement of the rotational force acting around an axis.

Understanding this unit is vital since it ties together the concepts of force and distance, showing how they work in tandem to produce torque. For our spark plug problem, the torque value is clearly defined as 15 Nm, indicating a fairly moderate amount of rotational force needed.

To ensure accurate calculations, it's crucial to keep units consistent throughout. Convert any lengths to meters, so when you multiply by force (in Newtons), the resulting torque remains in Newton meters. This consistency helps avoid errors and ensures your calculations are physically meaningful.