Problem 88 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer. A sliding door moves along the line of vector \(\overset{\rightharpoonup} {PQ}\). If a force is applied to the door along a vector that isorthogonal to \(\overset{\rightharpoonup} {PQ}\), then no work is done.

Step-by-Step Solution

Answer

The statement given is true. When a force is applied along a vector orthogonal to the movement of the sliding door, the work performed is zero.

Step 1: Understand the norms of work in physics

When covering topics related to work and energy in physics, it's important to understand that work is defined as the transfer of energy. In physics, work is defined as the product of the force (F) applied to an object and the distance (d) moved by the object due to the applied force. Mathematically, it is represented as W = F * d * cos(θ), where θ represents the angle between the force and the direction of movement.

Step 2: Apply this understanding to the current scenario

In this case, the force is applied along a vector orthogonal to movement vector \( \overset{\rightharpoonup} {PQ} \). Applying force in an orthogonal direction to the vector mean that the angle formed is 90 degrees.

Step 3: Perform the Calculation

Substituting θ = 90° into the work formula \( W = F * d * cos(θ) \), we obtain \( W = F * d * cos(90°) \). From trigonometric values, we know that cos(90°) is 0. Therefore, the work done is 0.

Key Concepts

Vector Orthogonality

When we encounter the term orthogonal in physics, especially in the realm of vectors, we are referring to the right angle relationship between two vectors. To visualize this, imagine two arrows intersecting at a point, where one points directly north and the other points directly east. These two arrows (or vectors) are orthogonal to each other, meaning they are at a 90-degree angle.

Orthogonal vectors have a special property in the context of work. Because work in physics can be seen as how much a force contributes to the movement of an object in the direction of that force, if a force is applied perpendicular to the movement, it does no work on the object. This concept is crucial in the understanding of the problem's solution.

Energy Transfer

Energy transfer is a core idea behind the concept of work in physics. Work is the process through which energy is moved from one system to another. As we apply force on an object and it causes that object to move, we are essentially transferring kinetic energy to it.

To calculate the amount of work done, and thus the energy transferred, we need to consider not only the magnitude of the force and the distance over which it acts but also the direction. This is encapsulated in the formula for work, which includes a cosine function of the angle between the force vector and the direction of motion. In our textbook problem, the fact that the force vector is orthogonal (at a right angle) to the motion vector immediately tells us something important about the energy transfer – it's zero!

Trigonometric Values

Understanding trigonometric values is fundamental in physics and mathematics. Functions such as sine, cosine, and tangent relate the angles of a triangle to the lengths of its sides. In the context of calculating work, the cosine of the angle between the force vector and the direction of motion is particularly important.

From trigonometry, we know that the cosine of 0 degrees is 1, which indicates full alignment between the force and displacement vectors, and thus, the maximum work done. Conversely, the cosine of 90 degrees is 0, meaning that if the force is applied perpendicular to the displacement, cos(90°), it contributes no energy to the object's motion in the direction of the force. This trigonometric value is the key to understanding why, in the exercise problem, when the force is orthogonal to the motion, the work done on the sliding door is zero.