Problem 82 - Precalculus

Question

Torque. You walk through a swinging mall door to enter a department store. You exert a force of \(40 \mathrm{N}\) applied perpendicular to the door. The door is 0.85 meters wide. Assuming that you pushed the door at its edge and the hinge is the pivot point, find the magnitude of the torque.

Step-by-Step Solution

Answer

The torque is 34 Nm.

Step 1: Understand Torque Formula

The torque \( \tau \) produced by a force is given by the formula: \( \tau = F \times r \times \sin(\theta) \), where \( F \) is the force applied, \( r \) is the distance from the pivot point (length of the lever arm), and \( \theta \) is the angle between the force and the lever arm. In this problem, the force is applied perpendicular to the door, so \( \theta = 90^\circ \) and \( \sin(90^\circ) = 1 \).

Step 2: Substitute Known Values

Substitute the known values into the torque formula. We have \( F = 40 \mathrm{N} \), \( r = 0.85 \mathrm{m} \), and since the force is perpendicular, \( \sin(90^\circ) = 1 \). The formula becomes \( \tau = 40 \mathrm{N} \times 0.85 \mathrm{m} \times 1 \).

Step 3: Calculate the Torque

Perform the multiplication to find the torque: \( \tau = 40 \mathrm{N} \times 0.85 \mathrm{m} = 34 \mathrm{Nm} \).

Key Concepts

Force

In the realm of physics, force is a fundamental concept, representing a push or pull exerted on an object. In the context of the exercise, you exert a force of 40 Newtons on the door. This force acts as a vector quantity, meaning it has both magnitude and direction. For instance, you push the door forward to open it. Understanding force is crucial for calculating torque, as it directly influences the rotational effect produced. Newtons, the unit of measure for force, quantify the amount necessary to accelerate a mass at a certain rate. This unit combines mass and acceleration elements, showcasing the dynamic aspect of force.
To visualize, imagine you push a shopping cart. The harder you push, the greater the force applied. Similarly, when opening the door, the 40 Newtons of force initiates the rotation around the hinge, facilitating movement.

Lever Arm

The lever arm is another key player when dealing with torque. It is the distance between the pivot point and where the force is applied. In our scenario, the door's width of 0.85 meters determines the lever arm's length. Think of the lever arm as the body's reach, dictating how far you can apply your force from the pivot. This distance amplifies or diminishes torque depending on its length. The longer the lever arm, the greater the potential torque.
Visualize swinging a baseball bat. Gripping it at the end maximizes the lever arm, allowing more power for a home run swing. The same concept applies to the door; applying force at its edge, furthest from the hinge, maximizes the torque.

Angle of Application

The angle of application is crucial in understanding torque. It is the angle between the direction of the applied force and the lever arm. This exercise makes it easy by stating the force is perpendicular, giving us an angle of 90 degrees. At 90 degrees, the calculation of torque simplifies as the sine of 90 degrees is 1. This maximizes the effective component of force contributing to torque.
To grasp this better, imagine pushing a door at different angles. Pushing perpendicular yields maximum force-directed into moving the door. Conversely, pushing at a shallow angle requires more effort with less rotational effect; hence, it's less efficient in creating torque.

Pivot Point

The pivot point is the axis around which the object, in this case, the door, rotates. Here, the hinge serves as the pivot point. It's akin to the fulcrum in a see-saw, a stable point allowing for rotation. Understanding the pivot point is vital for torque calculations as it anchors the lever arm and helps define the system's rotational dynamics.
Picture a door swinging open and closed. The hinge remains fixed, allowing smooth and consistent motion. Without this stable pivot, the rotational movement would be chaotic, leading to a potentially ineffective opening mechanism. The pivot point grounds the system, providing a center around which torque operates efficiently.