Question

To turn a stubborn bolt, \(80 \mathrm{lb}\) of force is applied to a \(10 \mathrm{in}\) wrench in a confined space, where the direction of applied force makes a \(10^{\circ}\) angle with the wrench. How much torque is subsequently applied to the wrench?

Short answer

The torque applied is 139 in⋅lb.

Step-by-step solution

Understand the Concept of Torque

Torque is the measure of the force causing an object to rotate. It depends not only on the amount of force applied but also on how far from the pivot point (or axis) the force is applied and the angle at which it is applied.

Define the Torque Equation

The equation for torque, \( \tau \), is given by \( \tau = r \times F \times \sin(\theta) \), where \( r \) is the distance from the pivot point to where the force is applied, \( F \) is the force, and \( \theta \) is the angle between the force and the lever arm.

Identify Given Values

We are given the force \( F = 80 \mathrm{lb} \), lever arm length \( r = 10 \mathrm{in} \), and angle \( \theta = 10^{\circ} \). These values need to be plugged into the torque equation.

Calculate the Sine of the Angle

Find \( \sin(10^{\circ}) \). Using a calculator, \( \sin(10^{\circ}) \approx 0.1736 \).

Substitute Values into Torque Equation

Insert the given values and the sine we calculated into the torque equation: \( \tau = 10 \mathrm{in} \times 80 \mathrm{lb} \times 0.1736 \).

Perform the Calculation

Calculate the torque: \( \tau = 10 \times 80 \times 0.1736 = 139 \mathrm{in} \cdot \mathrm{lb} \).

Conclusion of Torque Calculation

The torque applied to the wrench is \( 139 \mathrm{in} \cdot \mathrm{lb} \).

Key concepts

Force and Angle

When dealing with torque, understanding the role of force and angle is crucial. Torque is essentially a rotational force, which means it's directed in circles. Imagine trying to open a door: the force you apply with your hand helps the door turn around its hinges. The angle at which you apply this force can change how effectively the door swings.
- In our exercise, the 80 lb force applied is not directly perpendicular to the wrench. Instead, it forms a 10° angle with it. This angle affects how much the wrench turns when the force is applied.
- By knowing how angles influence torque, you can better understand mechanical systems and design them for optimal performance.
- If you apply the force at a lower angle, closer to zero, you are not using it as effectively to rotate the object. Conversely, at 90 degrees, it's fully effective.
The key takeaway here is that both the magnitude of the force and the angle at which it is applied determine the effectiveness of the resulting torque.

Torque Equation

The torque equation helps us quantify rotational force. It is given by \[\tau = r \times F \times \sin(\theta)\]Where:

• \(\tau\) is the torque.
• \(r\) is the distance from the pivot (lever arm length).
• \(F\) represents the force applied.
• \(\theta\) is the angle between the force and the lever arm.

In our example, we're dealing with a force of 80 lb, a lever arm of 10 in, and an angle of 10°.
- Using the formula, torque can be calculated by substituting these numbers into the equation.
- This requires calculating the sine of the angle, which adjusts the force based on its directionality.
- The formula illustrates how both distance and force, when combined with the correct angle, produce torque that turns or spins objects.

Lever Arm

The lever arm plays a significant role in the calculation of torque. It's the distance from the pivot point to where the force is applied. This distance is crucial because it acts as the 'handle' by which the force turns the object.
- In simpler terms, think of the lever arm as a broom handle: the longer the handle, the easier it is to sweep because you're applying force at a greater distance.
- In this exercise, the lever arm is the 10-inch wrench. This is the distance from the bolt (pivot) where the force is applied to the point of force application.
- Consequently, the greater this distance, the less force needed to achieve the same rotational effect, increasing the power of the torque.
The longer the lever arm, the more efficient the work per unit of force—making tasks easier and less effort-intensive. Using longer lever arms allows us to maximize our mechanical advantage.