Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Due to the scampering of a goat, a rock has been dislodged from a mountain and s sliding down an incline making a \(70^{\circ}\) angle with level ground. The weight of the rock exerts a downward force of 3 pounds. What is the component of this force in the direction of motion of the rock?

Step-by-Step Solution

Answer

The component of the force in the direction of the rock's motion is approximately 2.82 pounds. This is calculated by multiplying the rock's weight by the sine of the incline's angle.

Step 1: Understanding the Problem

The rock is dislodged and it slides down an incline making a \(70^{\circ}\) angle with the ground. The weight of the rock is 3 pounds. The weight acting downwards is the force causing the rock to move down the incline. We need to find the component of this force in the direction of the rock's motion, i.e., along the incline.

Step 2: Using Trigonometry to Decompose the Force

The force along the incline (or the component of the force in the direction of the rock’s slide down the incline) can be given as \(F_{x} = F \cdot \sin(\theta)\), where \(F\) is the force exerted and \(\theta\) is the angle of the incline with the ground. Here, \(F = 3\) pounds (weight of the rock) and \(\theta = 70^\circ\).

Step 3: Calculating the Component of Force

Using the formula from step 2, we now calculate the component of force as : \(F_{x} = F \cdot \sin(\theta) = 3 \cdot \sin(70^\circ)\). This can be calculated using a calculator where the sine function is available.

Key Concepts

Force Components

In physics, understanding force components is crucial, especially when analyzing the impact of forces in different directions. Imagine a rock sliding downhill. The rock's total weight acts directly downward, but not all this force helps it slide downward effectively.
To fully grasp force components, consider the weight acting on the rock, depicted by a vector pointing straight down. This vector can be broken into two parts:

• The force contributing to movement along the inclined plane.
• The force acting perpendicular to this plane.

This breakdown allows us to focus on just the force driving the rock down the slope. Using trigonometric functions like sine and cosine, we can find these components. In this topic, since we need the force along the slope, we use the sine of the angle of inclination. It's a handy approach that simplifies complex problems involving forces.

Inclined Plane

An inclined plane is a flat surface tilted at an angle to the horizontal. It's a common tool used to ease the movement of heavy objects, which slide or roll down under the influence of gravity. The angle at which this plane is inclined is critical as it determines how forces are distributed.
In our example, the 70-degree angle of the mountain's incline is significant. It tells us how steep the slope is and helps us calculate the force component that actually drives the rock downward. The steeper the plane, the larger the component of gravitational force that propels the rock.
Inclined planes are beneficial as they demonstrate real-life applications of trigonometry and physics. They show how objects can move more easily under the influence of gravity when assisted by a slope. Through these concepts, we understand movement and forces in mechanics better.

Decomposition of Forces

The decomposition of forces is a fundamental concept in physics that involves breaking down a force into its effective components. It allows us to better understand how different factors influence the movement of objects.
In the scenario with the falling rock, we decompose the gravitational force acting on the rock into two parts:

• The force along the inclined plane (the one we need to calculate).
• The force perpendicular to the plane, which doesn't affect the sliding motion.

To achieve this, we use trigonometric functions, specifically sine, to find the desired force component. The formula used is:
\[ F_{x} = F imes \sin(\theta) \]where \( F \) represents the rock's weight and \( \theta \) the plane's inclination angle.
This process simplifies complex motion analyses, revealing how much force is applied in specific directions. It's invaluable in physics and engineering, making calculations more accessible and results more applicable.