Question

Suppose you apply a force of 1.5 lb in a direction perpendicular to the screen of a laptop at a distance of 10 in from the hinge of the screen. Find the magnitude of torque (in \(\mathrm{ft} \cdot\) lb) that you apply.

Short answer

Answer: The magnitude of the torque produced is 1.25 ft-lb.

Step-by-step solution

Convert the distance to feet

Since there are 12 inches in 1 foot, we can divide the distance by 12 to convert it to feet: \(r = \frac{10}{12} = \frac{5}{6}\:\text{ft}\).

Calculate the force components in the direction of the torque calculation

As the force is applied perpendicularly to the laptop screen, we have \(\theta = 90^{\circ}\), and therefore, \(\sin\theta = \sin{90^{\circ}} = 1\). So, there's no need to calculate the force components.

Calculate the torque

Now that we have the distance in feet and the force applied, we can use the torque formula to find the magnitude of the torque: \(\tau = rf\sin\theta = \frac{5}{6}\:\text{ft} \times 1.5\:\text{lb} \times 1 = 1.25\:\text{ft}\cdot\text{lb}\). So, the magnitude of the torque applied is 1.25 ft-lb.

Key concepts

Understanding Force

Force is a fundamental concept in physics, and it refers to any interaction that, when unopposed, will change the motion of an object. In simple terms, a force can either push or pull an object. The unit of force in the imperial system is pounds (lb), and it's important to understand that force can have both magnitude and direction. In the given exercise, you apply a force of 1.5 lb, which is a measure of how much you are pushing against the laptop screen. Since the force is perpendicular to the screen, it maximizes its effect for calculating torque, because the sine of 90 degrees is 1. This maximization occurs because there is no need for any additional component calculations, making the torque computation straightforward.

Distance Conversion in Physics

When solving physics problems, it's essential to convert all measurements into consistent units. This process often involves converting distances from inches to feet, because most physics calculations, like this torque problem, prefer one common unit.

• 12 inches make up 1 foot.
• To convert inches to feet, divide the number of inches by 12.

In this exercise, the initial distance is 10 inches. By dividing 10 by 12, we get \(\frac{5}{6}\) feet. This conversion ensures that all units can work together neatly in the torque formula, providing an accurate and efficient computation.

Role of Trigonometry in Torque Calculation

Trigonometry plays a crucial role in physics, especially for problems that involve angles, like torque calculations. Torque (\(\tau\)) is determined using the formula:\[\tau = rF\sin\theta\]where \(r\) is the lever arm distance, \(F\) is the force applied, and \(\theta\) is the angle between the force and lever arm.
In this problem:

• The angle \(\theta\) is 90 degrees.
• The sine of 90 degrees is 1, which simplifies calculations as it indicates maximum torque efficiency.

Understanding the trigonometric function \(\sin\theta\) is vital because it determines the effective magnitude of the applied force in the intended direction. This makes torque problems easier when we're dealing with right angles.

Physics Problem Solving

Physics problem solving involves a structured approach to simplify and solve various real-world phenomena mathematically. Let’s explore the steps used in the given exercise:

• **Identify the problem elements** - Recognize the key variables: distance, force, and angle.
• **Convert units where necessary** - Use consistent units across all variables to maintain accuracy in calculations.
• **Apply relevant equations** - Torque = distance x force x \(\sin\theta\).
• **Solve step-by-step** - Break down the problem, convert units, plug values into equations, and simplify results.

By following these steps diligently, we transformed the real-world application of closing a laptop into a physics problem, offering insight into how these concepts manifest in everyday life. This structured problem-solving methodology can be applied to a multitude of physics scenarios.