(II) The bolts on the cylinder head of an engine require tightening to a torque of 95 m N. If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?

Verified step by step guidance

1. Step 1: Understand the relationship between torque, force, and lever arm length. Torque (τ) is given by the formula: $\tau =F\times r\times \sin \left(\theta \right)$, where F is the force applied, r is the length of the lever arm, and θ is the angle between the force and the lever arm. In this case, the force is applied perpendicular to the wrench, so $\sin \left(\theta \right)=1$.
2. Step 2: Convert the length of the wrench from centimeters to meters to ensure consistent units. Since 1 meter = 100 centimeters, the length of the wrench is $28/100=0.28$ meters.
3. Step 3: Rearrange the torque formula to solve for the force (F). The formula becomes: $F=\tau /r$, since the angle is 90 degrees and $\sin \left(90\right)=1$.
4. Step 4: Substitute the given values into the formula. The torque (τ) is 95 m·N, and the lever arm length (r) is 0.28 meters. The equation becomes: $F=95/0.28$.
5. Step 5: Perform the division to find the force (F). This will give the magnitude of the force the mechanic must exert perpendicular to the wrench at its end.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). It is expressed in units such as Newton-meters (N·m). In this context, the required torque of 95 m·N indicates the amount of rotational force needed to tighten the bolts.

Lever Arm

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. In this problem, the length of the wrench (28 cm) acts as the lever arm, which amplifies the force applied by the mechanic. A longer lever arm allows for greater torque with less force.

Force Calculation

To find the force exerted at the end of the wrench, the relationship between torque, force, and lever arm is used: Torque = Force × Lever Arm. Rearranging this formula allows us to calculate the necessary force by dividing the torque by the length of the wrench. This calculation is essential for determining how much force the mechanic needs to apply.