Question

Which is greater? Which has a greater horizontal component, a 100-N force directed at an angle of \(60^{\circ}\) above the horizontal or a 60-N force directed at an angle of \(30^{\circ}\) above the horizontal?

Short answer

Answer: The 60-N force directed at an angle of 30° above the horizontal has the greater horizontal component.

Step-by-step solution

Understand the given forces

A 100-N force is directed at an angle of 60° above the horizontal (Force F1), and a 60-N force is directed at an angle of 30° above the horizontal (Force F2).

Determine the horizontal components

The horizontal component of a force can be found using the trigonometric function cosine, in the following formula: \(\text{Horizontal Component} = \text{Force Magnitude} \times \cos(\text{Angle})\)

Calculate the horizontal component of Force F1

Using the formula mentioned above, the horizontal component of F1 is: \(F_{1x} = F_{1}\cos\theta_{1} = 100 \times \cos(60^{\circ})\) \(F_{1x} = 100 \times \frac{1}{2} = 50 \, \text{N}\)

Calculate the horizontal component of Force F2

Using the same formula, the horizontal component of F2 is: \(F_{2x} = F_{2}\cos\theta_{2} = 60 \times \cos(30^{\circ})\) \(F_{2x} = 60 \times \frac{\sqrt{3}}{2} = 30\sqrt{3} \, \text{N}\)

Compare the horizontal components

Now, we need to compare the magnitudes of \(F_{1x}\) and \(F_{2x}\) to determine which force has the greater horizontal component. - \(F_{1x} = 50 \, \text{N}\) - \(F_{2x} = 30\sqrt{3} \, \text{N} \approx 51.96 \, \text{N}\) As \(F_{2x} > F_{1x}\), the 60-N force directed at an angle of 30° above the horizontal has the greater horizontal component.

Key concepts

Trigonometric Functions

Trigonometric functions play a crucial role in understanding and analyzing different components of forces, especially when dealing with angles. In this context, the cosine function is particularly important as it helps us determine the horizontal component of a force. The basic trigonometric functions are sine, cosine, and tangent, represented as sin, cos, and tan, respectively. Cosine, in particular, measures the adjacent side over the hypotenuse in a right-angled triangle. Whenever you have a force at an angle, you can use the cosine function to split the force into its horizontal component.
Here is the formula:

• Horizontal Component = Force Magnitude × cos(Angle)

Understanding how to use trigonometric functions is essential in physics for breaking down forces into components, which simplifies problem-solving and force analysis.

Horizontal Components

When analyzing forces, it's often necessary to understand the individual components that make up a force vector. The horizontal component is one such important part that represents how much of the force acts along a horizontal plane. It effectively describes how much of a push or pull there is in the horizontal direction. To find the horizontal component, you need two key pieces of information: the force magnitude and the angle of the force relative to the horizontal axis.
Using the cosine function, we calculate this component. For example, the horizontal component for a 100-N force at a 60-degree angle is calculated using:\( F_{1x} = F_{1}\cos\theta_{1} = 100 \times \cos(60^{\circ}) = 50 \text{ N} \)Similarly, for a 60-N force at a 30-degree angle:\( F_{2x} = F_{2}\cos\theta_{2} = 60 \times \cos(30^{\circ}) = 30\sqrt{3} \text{ N} \)
This ability to decompose forces into horizontal components is pivotal in many fields, including engineering and physics, as it allows for a clearer understanding of how different elements interact within a system.

Force Analysis

Force analysis involves breaking down complex forces into simpler, analyzable components to understand their effects better. By examining each component, we gain insights into how a force operates within a system. In this exercise, we focused on analyzing the horizontal components of two forces to determine which one exerted a greater influence in the horizontal direction.
To carry out a force analysis:

• Identify each force and its direction.
• Use trigonometric functions to break forces into components.
• Compare the components to see how they influence the system.

In this case, by computing the horizontal components, it was determined that the 60-N force directed at a 30-degree angle had a greater horizontal component (\( \approx 51.96 \text{ N} \)) compared to the 100-N force at a 60-degree angle (\( 50 \text{ N} \)).
This kind of analysis is not only useful in theoretical physics problems but also practical applications like engineering design, where understanding forces is crucial for building structures that can withstand various loads and stresses. By mastering force analysis, one can predict and manipulate how different forces will act in real-world scenarios.